Precalculus Examples

$f (x, y) = \ln (9 - x^{2} - 9 y^{2})$

Step 1

Subtract $\ln (9 - x^{2} - 9 y^{2})$ from both sides of the equation.

$f (x, y) - \ln (9 - x^{2} - 9 y^{2}) = 0$

Step 2

To solve for $y$ , rewrite the equation using properties of logarithms.

$e^{\ln (9 - x^{2} - 9 y^{2})} = e^{f (x, y)}$

Step 3

Rewrite $\ln (9 - x^{2} - 9 y^{2}) = f (x, y)$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$e^{f (x, y)} = 9 - x^{2} - 9 y^{2}$

Step 4

Solve for $y$ .

Tap for more steps...

Step 4.1

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (e^{f (x, y)}) = \ln (9 - x^{2} - 9 y^{2})$

Step 4.2

Expand the left side.

Tap for more steps...

Step 4.2.1

Expand $\ln (e^{f (x, y)})$ by moving $f (x, y)$ outside the logarithm.

$f (x, y) \ln (e) = \ln (9 - x^{2} - 9 y^{2})$

Step 4.2.2

The natural logarithm of $e$ is $1$ .

$f (x, y) \cdot 1 = \ln (9 - x^{2} - 9 y^{2})$

Step 4.2.3

Multiply $f$ by $1$ .

$f (x, y) = \ln (9 - x^{2} - 9 y^{2})$

Step 4.3

Subtract $\ln (9 - x^{2} - 9 y^{2})$ from both sides of the equation.

$f (x, y) - \ln (9 - x^{2} - 9 y^{2}) = 0$

Step 4.4

To solve for $y$ , rewrite the equation using properties of logarithms.

$e^{\ln (9 - x^{2} - 9 y^{2})} = e^{f (x, y)}$

Step 4.5

$e^{f (x, y)} = 9 - x^{2} - 9 y^{2}$

Step 4.6

Solve for $y$ .

Tap for more steps...

Step 4.6.1

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (e^{f (x, y)}) = \ln (9 - x^{2} - 9 y^{2})$

Step 4.6.2

Expand the left side.

Tap for more steps...

Step 4.6.2.1

Expand $\ln (e^{f (x, y)})$ by moving $f (x, y)$ outside the logarithm.

$f (x, y) \ln (e) = \ln (9 - x^{2} - 9 y^{2})$

Step 4.6.2.2

The natural logarithm of $e$ is $1$ .

$f (x, y) \cdot 1 = \ln (9 - x^{2} - 9 y^{2})$

Step 4.6.2.3

Multiply $f$ by $1$ .

$f (x, y) = \ln (9 - x^{2} - 9 y^{2})$

Step 4.6.3

Subtract $\ln (9 - x^{2} - 9 y^{2})$ from both sides of the equation.

$f (x, y) - \ln (9 - x^{2} - 9 y^{2}) = 0$

Step 4.6.4

To solve for $y$ , rewrite the equation using properties of logarithms.

$e^{\ln (9 - x^{2} - 9 y^{2})} = e^{f (x, y)}$

Step 4.6.5

$e^{f (x, y)} = 9 - x^{2} - 9 y^{2}$

Step 4.6.6

Solve for $y$ .

Tap for more steps...

Step 4.6.6.1

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (e^{f (x, y)}) = \ln (9 - x^{2} - 9 y^{2})$

Step 4.6.6.2

Expand the left side.

Tap for more steps...

Step 4.6.6.2.1

Expand $\ln (e^{f (x, y)})$ by moving $f (x, y)$ outside the logarithm.

$f (x, y) \ln (e) = \ln (9 - x^{2} - 9 y^{2})$

Step 4.6.6.2.2

The natural logarithm of $e$ is $1$ .

$f (x, y) \cdot 1 = \ln (9 - x^{2} - 9 y^{2})$

Step 4.6.6.2.3

Multiply $f$ by $1$ .

$f (x, y) = \ln (9 - x^{2} - 9 y^{2})$

Step 4.6.6.3

Subtract $\ln (9 - x^{2} - 9 y^{2})$ from both sides of the equation.

$f (x, y) - \ln (9 - x^{2} - 9 y^{2}) = 0$

Step 4.6.6.4

To solve for $y$ , rewrite the equation using properties of logarithms.

$e^{\ln (9 - x^{2} - 9 y^{2})} = e^{f (x, y)}$

Step 4.6.6.5

$e^{f (x, y)} = 9 - x^{2} - 9 y^{2}$

Step 4.6.6.6

Solve for $y$ .

Tap for more steps...

Step 4.6.6.6.1

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (e^{f (x, y)}) = \ln (9 - x^{2} - 9 y^{2})$

Step 4.6.6.6.2

Expand the left side.

Tap for more steps...

Step 4.6.6.6.2.1

Expand $\ln (e^{f (x, y)})$ by moving $f (x, y)$ outside the logarithm.

$f (x, y) \ln (e) = \ln (9 - x^{2} - 9 y^{2})$

Step 4.6.6.6.2.2

The natural logarithm of $e$ is $1$ .

$f (x, y) \cdot 1 = \ln (9 - x^{2} - 9 y^{2})$

Step 4.6.6.6.2.3

Multiply $f$ by $1$ .

$f (x, y) = \ln (9 - x^{2} - 9 y^{2})$

Step 5

Set the argument in $\ln (9 - x^{2} - 9 y^{2})$ greater than $0$ to find where the expression is defined.

$9 - x^{2} - 9 y^{2} > 0$

Step 6

Solve for $x$ .

Tap for more steps...

Step 6.1

Move all terms not containing $x$ to the right side of the inequality.

Tap for more steps...

Step 6.1.1

Subtract $9$ from both sides of the inequality.

$- x^{2} - 9 y^{2} > - 9$

Step 6.1.2

Add $9 y^{2}$ to both sides of the inequality.

$- x^{2} > - 9 + 9 y^{2}$

Step 6.2

Divide each term in $- x^{2} > - 9 + 9 y^{2}$ by $- 1$ and simplify.

Tap for more steps...

Step 6.2.1

Divide each term in $- x^{2} > - 9 + 9 y^{2}$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x^{2}}{- 1} < \frac{- 9}{- 1} + \frac{9 y^{2}}{- 1}$

Step 6.2.2

Simplify the left side.

Tap for more steps...

Step 6.2.2.1

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} < \frac{- 9}{- 1} + \frac{9 y^{2}}{- 1}$

Step 6.2.2.2

Divide $x^{2}$ by $1$ .

$x^{2} < \frac{- 9}{- 1} + \frac{9 y^{2}}{- 1}$

Step 6.2.3

Simplify the right side.

Tap for more steps...

Step 6.2.3.1

Simplify each term.

Tap for more steps...

Step 6.2.3.1.1

Divide $- 9$ by $- 1$ .

$x^{2} < 9 + \frac{9 y^{2}}{- 1}$

Step 6.2.3.1.2

Move the negative one from the denominator of $\frac{9 y^{2}}{- 1}$ .

$x^{2} < 9 - 1 \cdot (9 y^{2})$

Step 6.2.3.1.3

Rewrite $- 1 \cdot (9 y^{2})$ as $- (9 y^{2})$ .

$x^{2} < 9 - (9 y^{2})$

Step 6.2.3.1.4

Multiply $9$ by $- 1$ .

$x^{2} < 9 - 9 y^{2}$

Step 6.3

Take the specified root of both sides of the inequality to eliminate the exponent on the left side.

$\sqrt{x^{2}} < \sqrt{9 - 9 y^{2}}$

Step 6.4

Simplify the equation.

Tap for more steps...

Step 6.4.1

Simplify the left side.

Tap for more steps...

Step 6.4.1.1

Pull terms out from under the radical.

$| x | < \sqrt{9 - 9 y^{2}}$

Step 6.4.2

Simplify the right side.

Tap for more steps...

Step 6.4.2.1

Simplify $\sqrt{9 - 9 y^{2}}$ .

Tap for more steps...

Step 6.4.2.1.1

Factor $9$ out of $9 - 9 y^{2}$ .

Tap for more steps...

Step 6.4.2.1.1.1

Factor $9$ out of $9$ .

$| x | < \sqrt{9 (1) - 9 y^{2}}$

Step 6.4.2.1.1.2

Factor $9$ out of $- 9 y^{2}$ .

$| x | < \sqrt{9 (1) + 9 (- y^{2})}$

Step 6.4.2.1.1.3

Factor $9$ out of $9 (1) + 9 (- y^{2})$ .

$| x | < \sqrt{9 (1 - y^{2})}$

Step 6.4.2.1.2

Rewrite $1$ as $1^{2}$ .

$| x | < \sqrt{9 (1^{2} - y^{2})}$

Step 6.4.2.1.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = y$ .

$| x | < \sqrt{9 (1 + y) (1 - y)}$

Step 6.4.2.1.4

Rewrite $9 (1 + y) (1 - y)$ as $3^{2} ((1^{2} + y) (1 - y))$ .

Tap for more steps...

Step 6.4.2.1.4.1

Rewrite $9$ as $3^{2}$ .

$| x | < \sqrt{3^{2} (1 + y) (1 - y)}$

Step 6.4.2.1.4.2

Rewrite $1$ as $1^{2}$ .

$| x | < \sqrt{3^{2} (1^{2} + y) (1 - y)}$

Step 6.4.2.1.4.3

Add parentheses.

$| x | < \sqrt{3^{2} ((1^{2} + y) (1 - y))}$

Step 6.4.2.1.5

Pull terms out from under the radical.

$| x | < | 3 | \sqrt{(1^{2} + y) (1 - y)}$

Step 6.4.2.1.6

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

$| x | < 3 \sqrt{(1^{2} + y) (1 - y)}$

Step 6.4.2.1.7

One to any power is one.

$| x | < 3 \sqrt{(1 + y) (1 - y)}$

Step 6.5

Write $| x | < 3 \sqrt{(1 + y) (1 - y)}$ as a piecewise.

Tap for more steps...

Step 6.5.1

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$x \geq 0$

Step 6.5.2

In the piece where $x$ is non-negative, remove the absolute value.

$x < 3 \sqrt{(1 + y) (1 - y)}$

Step 6.5.3

Find the domain of $x < 3 \sqrt{(1 + y) (1 - y)}$ and find the intersection with $x \geq 0$ .

Tap for more steps...

Step 6.5.3.1

Find the domain of $x < 3 \sqrt{(1 + y) (1 - y)}$ .

Tap for more steps...

Step 6.5.3.1.1

Set the radicand in $\sqrt{(1 + y) (1 - y)}$ greater than or equal to $0$ to find where the expression is defined.

$(1 + y) (1 - y) \geq 0$

Step 6.5.3.1.2

Solve for $x$ .

Tap for more steps...

Step 6.5.3.1.2.1

Simplify $(1 + y) (1 - y)$ .

Tap for more steps...

Step 6.5.3.1.2.1.1

Expand $(1 + y) (1 - y)$ using the FOIL Method.

Tap for more steps...

Step 6.5.3.1.2.1.1.1

Apply the distributive property.

$1 (1 - y) + y (1 - y) \geq 0$

Step 6.5.3.1.2.1.1.2

Apply the distributive property.

$1 \cdot 1 + 1 (- y) + y (1 - y) \geq 0$

Step 6.5.3.1.2.1.1.3

Apply the distributive property.

$1 \cdot 1 + 1 (- y) + y \cdot 1 + y (- y) \geq 0$

Step 6.5.3.1.2.1.2

Simplify and combine like terms.

Tap for more steps...

Step 6.5.3.1.2.1.2.1

Simplify each term.

Tap for more steps...

Step 6.5.3.1.2.1.2.1.1

Multiply $1$ by $1$ .

$1 + 1 (- y) + y \cdot 1 + y (- y) \geq 0$

Step 6.5.3.1.2.1.2.1.2

Multiply $- y$ by $1$ .

$1 - y + y \cdot 1 + y (- y) \geq 0$

Step 6.5.3.1.2.1.2.1.3

Multiply $y$ by $1$ .

$1 - y + y + y (- y) \geq 0$

Step 6.5.3.1.2.1.2.1.4

Rewrite using the commutative property of multiplication.

$1 - y + y - y \cdot y \geq 0$

Step 6.5.3.1.2.1.2.1.5

Multiply $y$ by $y$ by adding the exponents.

Tap for more steps...

Step 6.5.3.1.2.1.2.1.5.1

Move $y$ .

$1 - y + y - (y \cdot y) \geq 0$

Step 6.5.3.1.2.1.2.1.5.2

Multiply $y$ by $y$ .

$1 - y + y - y^{2} \geq 0$

Step 6.5.3.1.2.1.2.2

Add $- y$ and $y$ .

$1 + 0 - y^{2} \geq 0$

Step 6.5.3.1.2.1.2.3

Add $1$ and $0$ .

$1 - y^{2} \geq 0$

Step 6.5.3.1.2.2

Subtract $1$ from both sides of the inequality.

$- y^{2} \geq - 1$

Step 6.5.3.1.2.3

Divide each term in $- y^{2} \geq - 1$ by $- 1$ and simplify.

Tap for more steps...

Step 6.5.3.1.2.3.1

Divide each term in $- y^{2} \geq - 1$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- y^{2}}{- 1} \leq \frac{- 1}{- 1}$

Step 6.5.3.1.2.3.2

Simplify the left side.

Tap for more steps...

Step 6.5.3.1.2.3.2.1

Dividing two negative values results in a positive value.

$\frac{y^{2}}{1} \leq \frac{- 1}{- 1}$

Step 6.5.3.1.2.3.2.2

Divide $y^{2}$ by $1$ .

$y^{2} \leq \frac{- 1}{- 1}$

Step 6.5.3.1.2.3.3

Simplify the right side.

Tap for more steps...

Step 6.5.3.1.2.3.3.1

Divide $- 1$ by $- 1$ .

$y^{2} \leq 1$

Step 6.5.3.1.2.4

Take the specified root of both sides of the inequality to eliminate the exponent on the left side.

$\sqrt{y^{2}} \leq \sqrt{1}$

Step 6.5.3.1.2.5

Simplify the equation.

Tap for more steps...

Step 6.5.3.1.2.5.1

Simplify the left side.

Tap for more steps...

Step 6.5.3.1.2.5.1.1

Pull terms out from under the radical.

$| y | \leq \sqrt{1}$

Step 6.5.3.1.2.5.2

Simplify the right side.

Tap for more steps...

Step 6.5.3.1.2.5.2.1

Any root of $1$ is $1$ .

$| y | \leq 1$

Step 6.5.3.1.2.6

Write $| y | \leq 1$ as a piecewise.

Tap for more steps...

Step 6.5.3.1.2.6.1

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$y \geq 0$

Step 6.5.3.1.2.6.2

In the piece where $y$ is non-negative, remove the absolute value.

$y \leq 1$

Step 6.5.3.1.2.6.3

To find the interval for the second piece, find where the inside of the absolute value is negative.

$y < 0$

Step 6.5.3.1.2.6.4

In the piece where $y$ is negative, remove the absolute value and multiply by $- 1$ .

$- y \leq 1$

Step 6.5.3.1.2.6.5

Write as a piecewise.

${\begin{cases} y \leq 1 & y \geq 0 \\ - y \leq 1 & y < 0 \end{cases}$

Step 6.5.3.1.2.7

Find the intersection of $y \leq 1$ and $y \geq 0$ .

$0 \leq y \leq 1$

Step 6.5.3.1.2.8

Solve $- y \leq 1$ when $y < 0$ .

Tap for more steps...

Step 6.5.3.1.2.8.1

Divide each term in $- y \leq 1$ by $- 1$ and simplify.

Tap for more steps...

Step 6.5.3.1.2.8.1.1

Divide each term in $- y \leq 1$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- y}{- 1} \geq \frac{1}{- 1}$

Step 6.5.3.1.2.8.1.2

Simplify the left side.

Tap for more steps...

Step 6.5.3.1.2.8.1.2.1

Dividing two negative values results in a positive value.

$\frac{y}{1} \geq \frac{1}{- 1}$

Step 6.5.3.1.2.8.1.2.2

Divide $y$ by $1$ .

$y \geq \frac{1}{- 1}$

Step 6.5.3.1.2.8.1.3

Simplify the right side.

Tap for more steps...

Step 6.5.3.1.2.8.1.3.1

Divide $1$ by $- 1$ .

$y \geq - 1$

Step 6.5.3.1.2.8.2

Find the intersection of $y \geq - 1$ and $y < 0$ .

$- 1 \leq y < 0$

Step 6.5.3.1.2.9

Find the union of the solutions.

$- 1 \leq y \leq 1$

Step 6.5.3.1.3

The domain is all values of $x$ that make the expression defined.

$[- 1, 1]$

Step 6.5.3.2

Find the intersection of $x \geq 0$ and $[- 1, 1]$ .

$0 \leq x \leq 1$

Step 6.5.4

To find the interval for the second piece, find where the inside of the absolute value is negative.

$x < 0$

Step 6.5.5

In the piece where $x$ is negative, remove the absolute value and multiply by $- 1$ .

$- x < 3 \sqrt{(1 + y) (1 - y)}$

Step 6.5.6

Find the domain of $- x < 3 \sqrt{(1 + y) (1 - y)}$ and find the intersection with $x < 0$ .

Tap for more steps...

Step 6.5.6.1

Find the domain of $- x < 3 \sqrt{(1 + y) (1 - y)}$ .

Tap for more steps...

Step 6.5.6.1.1

Set the radicand in $\sqrt{(1 + y) (1 - y)}$ greater than or equal to $0$ to find where the expression is defined.

$(1 + y) (1 - y) \geq 0$

Step 6.5.6.1.2

Solve for $x$ .

Tap for more steps...

Step 6.5.6.1.2.1

Simplify $(1 + y) (1 - y)$ .

Tap for more steps...

Step 6.5.6.1.2.1.1

Expand $(1 + y) (1 - y)$ using the FOIL Method.

Tap for more steps...

Step 6.5.6.1.2.1.1.1

Apply the distributive property.

$1 (1 - y) + y (1 - y) \geq 0$

Step 6.5.6.1.2.1.1.2

Apply the distributive property.

$1 \cdot 1 + 1 (- y) + y (1 - y) \geq 0$

Step 6.5.6.1.2.1.1.3

Apply the distributive property.

$1 \cdot 1 + 1 (- y) + y \cdot 1 + y (- y) \geq 0$

Step 6.5.6.1.2.1.2

Simplify and combine like terms.

Tap for more steps...

Step 6.5.6.1.2.1.2.1

Simplify each term.

Tap for more steps...

Step 6.5.6.1.2.1.2.1.1

Multiply $1$ by $1$ .

$1 + 1 (- y) + y \cdot 1 + y (- y) \geq 0$

Step 6.5.6.1.2.1.2.1.2

Multiply $- y$ by $1$ .

$1 - y + y \cdot 1 + y (- y) \geq 0$

Step 6.5.6.1.2.1.2.1.3

Multiply $y$ by $1$ .

$1 - y + y + y (- y) \geq 0$

Step 6.5.6.1.2.1.2.1.4

Rewrite using the commutative property of multiplication.

$1 - y + y - y \cdot y \geq 0$

Step 6.5.6.1.2.1.2.1.5

Multiply $y$ by $y$ by adding the exponents.

Tap for more steps...

Step 6.5.6.1.2.1.2.1.5.1

Move $y$ .

$1 - y + y - (y \cdot y) \geq 0$

Step 6.5.6.1.2.1.2.1.5.2

Multiply $y$ by $y$ .

$1 - y + y - y^{2} \geq 0$

Step 6.5.6.1.2.1.2.2

Add $- y$ and $y$ .

$1 + 0 - y^{2} \geq 0$

Step 6.5.6.1.2.1.2.3

Add $1$ and $0$ .

$1 - y^{2} \geq 0$

Step 6.5.6.1.2.2

Subtract $1$ from both sides of the inequality.

$- y^{2} \geq - 1$

Step 6.5.6.1.2.3

Divide each term in $- y^{2} \geq - 1$ by $- 1$ and simplify.

Tap for more steps...

Step 6.5.6.1.2.3.1

Divide each term in $- y^{2} \geq - 1$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- y^{2}}{- 1} \leq \frac{- 1}{- 1}$

Step 6.5.6.1.2.3.2

Simplify the left side.

Tap for more steps...

Step 6.5.6.1.2.3.2.1

Dividing two negative values results in a positive value.

$\frac{y^{2}}{1} \leq \frac{- 1}{- 1}$

Step 6.5.6.1.2.3.2.2

Divide $y^{2}$ by $1$ .

$y^{2} \leq \frac{- 1}{- 1}$

Step 6.5.6.1.2.3.3

Simplify the right side.

Tap for more steps...

Step 6.5.6.1.2.3.3.1

Divide $- 1$ by $- 1$ .

$y^{2} \leq 1$

Step 6.5.6.1.2.4

Take the specified root of both sides of the inequality to eliminate the exponent on the left side.

$\sqrt{y^{2}} \leq \sqrt{1}$

Step 6.5.6.1.2.5

Simplify the equation.

Tap for more steps...

Step 6.5.6.1.2.5.1

Simplify the left side.

Tap for more steps...

Step 6.5.6.1.2.5.1.1

Pull terms out from under the radical.

$| y | \leq \sqrt{1}$

Step 6.5.6.1.2.5.2

Simplify the right side.

Tap for more steps...

Step 6.5.6.1.2.5.2.1

Any root of $1$ is $1$ .

$| y | \leq 1$

Step 6.5.6.1.2.6

Write $| y | \leq 1$ as a piecewise.

Tap for more steps...

Step 6.5.6.1.2.6.1

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$y \geq 0$

Step 6.5.6.1.2.6.2

In the piece where $y$ is non-negative, remove the absolute value.

$y \leq 1$

Step 6.5.6.1.2.6.3

To find the interval for the second piece, find where the inside of the absolute value is negative.

$y < 0$

Step 6.5.6.1.2.6.4

In the piece where $y$ is negative, remove the absolute value and multiply by $- 1$ .

$- y \leq 1$

Step 6.5.6.1.2.6.5

Write as a piecewise.

${\begin{cases} y \leq 1 & y \geq 0 \\ - y \leq 1 & y < 0 \end{cases}$

Step 6.5.6.1.2.7

Find the intersection of $y \leq 1$ and $y \geq 0$ .

$0 \leq y \leq 1$

Step 6.5.6.1.2.8

Solve $- y \leq 1$ when $y < 0$ .

Tap for more steps...

Step 6.5.6.1.2.8.1

Divide each term in $- y \leq 1$ by $- 1$ and simplify.

Tap for more steps...

Step 6.5.6.1.2.8.1.1

Divide each term in $- y \leq 1$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- y}{- 1} \geq \frac{1}{- 1}$

Step 6.5.6.1.2.8.1.2

Simplify the left side.

Tap for more steps...

Step 6.5.6.1.2.8.1.2.1

Dividing two negative values results in a positive value.

$\frac{y}{1} \geq \frac{1}{- 1}$

Step 6.5.6.1.2.8.1.2.2

Divide $y$ by $1$ .

$y \geq \frac{1}{- 1}$

Step 6.5.6.1.2.8.1.3

Simplify the right side.

Tap for more steps...

Step 6.5.6.1.2.8.1.3.1

Divide $1$ by $- 1$ .

$y \geq - 1$

Step 6.5.6.1.2.8.2

Find the intersection of $y \geq - 1$ and $y < 0$ .

$- 1 \leq y < 0$

Step 6.5.6.1.2.9

Find the union of the solutions.

$- 1 \leq y \leq 1$

Step 6.5.6.1.3

The domain is all values of $x$ that make the expression defined.

$[- 1, 1]$

Step 6.5.6.2

Find the intersection of $x < 0$ and $[- 1, 1]$ .

$- 1 \leq x < 0$

Step 6.5.7

Write as a piecewise.

${\begin{cases} x < 3 \sqrt{(1 + y) (1 - y)} & 0 \leq x \leq 1 \\ - x < 3 \sqrt{(1 + y) (1 - y)} & - 1 \leq x < 0 \end{cases}$

Step 6.6

Solve $x < 3 \sqrt{(1 + y) (1 - y)}$ when $0 \leq x \leq 1$ .

Tap for more steps...

Step 6.6.1

Solve $x < 3 \sqrt{(1 + y) (1 - y)}$ for $y$ .

Tap for more steps...

Step 6.6.1.1

Rewrite so $y$ is on the left side of the inequality.

$3 \sqrt{(1 + y) (1 - y)} > x$

Step 6.6.1.2

Divide each term in $3 \sqrt{(1 + y) (1 - y)} > x$ by $3$ and simplify.

Tap for more steps...

Step 6.6.1.2.1

Divide each term in $3 \sqrt{(1 + y) (1 - y)} > x$ by $3$ .

$\frac{3 \sqrt{(1 + y) (1 - y)}}{3} > \frac{x}{3}$

Step 6.6.1.2.2

Simplify the left side.

Tap for more steps...

Step 6.6.1.2.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 6.6.1.2.2.1.1

Cancel the common factor.

$\frac{3 \sqrt{(1 + y) (1 - y)}}{3} > \frac{x}{3}$

Step 6.6.1.2.2.1.2

Divide $\sqrt{(1 + y) (1 - y)}$ by $1$ .

$\sqrt{(1 + y) (1 - y)} > \frac{x}{3}$

Step 6.6.1.3

To remove the radical on the left side of the inequality, square both sides of the inequality.

${\sqrt{(1 + y) (1 - y)}}^{2} > {(\frac{x}{3})}^{2}$

Step 6.6.1.4

Simplify each side of the inequality.

Tap for more steps...

Step 6.6.1.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{(1 + y) (1 - y)}$ as ${((1 + y) (1 - y))}^{\frac{1}{2}}$ .

${({((1 + y) (1 - y))}^{\frac{1}{2}})}^{2} > {(\frac{x}{3})}^{2}$

Step 6.6.1.4.2

Simplify the left side.

Tap for more steps...

Step 6.6.1.4.2.1

Simplify ${({((1 + y) (1 - y))}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 6.6.1.4.2.1.1

Multiply the exponents in ${({((1 + y) (1 - y))}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 6.6.1.4.2.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${((1 + y) (1 - y))}^{\frac{1}{2} \cdot 2} > {(\frac{x}{3})}^{2}$

Step 6.6.1.4.2.1.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.6.1.4.2.1.1.2.1

Cancel the common factor.

${((1 + y) (1 - y))}^{\frac{1}{2} \cdot 2} > {(\frac{x}{3})}^{2}$

Step 6.6.1.4.2.1.1.2.2

Rewrite the expression.

${((1 + y) (1 - y))}^{1} > {(\frac{x}{3})}^{2}$

Step 6.6.1.4.2.1.2

Expand $(1 + y) (1 - y)$ using the FOIL Method.

Tap for more steps...

Step 6.6.1.4.2.1.2.1

Apply the distributive property.

${(1 (1 - y) + y (1 - y))}^{1} > {(\frac{x}{3})}^{2}$

Step 6.6.1.4.2.1.2.2

Apply the distributive property.

${(1 \cdot 1 + 1 (- y) + y (1 - y))}^{1} > {(\frac{x}{3})}^{2}$

Step 6.6.1.4.2.1.2.3

Apply the distributive property.

${(1 \cdot 1 + 1 (- y) + y \cdot 1 + y (- y))}^{1} > {(\frac{x}{3})}^{2}$

Step 6.6.1.4.2.1.3

Simplify and combine like terms.

Tap for more steps...

Step 6.6.1.4.2.1.3.1

Simplify each term.

Tap for more steps...

Step 6.6.1.4.2.1.3.1.1

Multiply $1$ by $1$ .

${(1 + 1 (- y) + y \cdot 1 + y (- y))}^{1} > {(\frac{x}{3})}^{2}$

Step 6.6.1.4.2.1.3.1.2

Multiply $- y$ by $1$ .

${(1 - y + y \cdot 1 + y (- y))}^{1} > {(\frac{x}{3})}^{2}$

Step 6.6.1.4.2.1.3.1.3

Multiply $y$ by $1$ .

${(1 - y + y + y (- y))}^{1} > {(\frac{x}{3})}^{2}$

Step 6.6.1.4.2.1.3.1.4

Rewrite using the commutative property of multiplication.

${(1 - y + y - y \cdot y)}^{1} > {(\frac{x}{3})}^{2}$

Step 6.6.1.4.2.1.3.1.5

Multiply $y$ by $y$ by adding the exponents.

Tap for more steps...

Step 6.6.1.4.2.1.3.1.5.1

Move $y$ .

${(1 - y + y - (y \cdot y))}^{1} > {(\frac{x}{3})}^{2}$

Step 6.6.1.4.2.1.3.1.5.2

Multiply $y$ by $y$ .

${(1 - y + y - y^{2})}^{1} > {(\frac{x}{3})}^{2}$

Step 6.6.1.4.2.1.3.2

Add $- y$ and $y$ .

${(1 + 0 - y^{2})}^{1} > {(\frac{x}{3})}^{2}$

Step 6.6.1.4.2.1.3.3

Add $1$ and $0$ .

${(1 - y^{2})}^{1} > {(\frac{x}{3})}^{2}$

Step 6.6.1.4.2.1.4

Simplify.

$1 - y^{2} > {(\frac{x}{3})}^{2}$

Step 6.6.1.4.3

Simplify the right side.

Tap for more steps...

Step 6.6.1.4.3.1

Simplify ${(\frac{x}{3})}^{2}$ .

Tap for more steps...

Step 6.6.1.4.3.1.1

Apply the product rule to $\frac{x}{3}$ .

$1 - y^{2} > \frac{x^{2}}{3^{2}}$

Step 6.6.1.4.3.1.2

Raise $3$ to the power of $2$ .

$1 - y^{2} > \frac{x^{2}}{9}$

Step 6.6.1.5

Solve for $y$ .

Tap for more steps...

Step 6.6.1.5.1

Subtract $1$ from both sides of the inequality.

$- y^{2} > \frac{x^{2}}{9} - 1$

Step 6.6.1.5.2

Divide each term in $- y^{2} > \frac{x^{2}}{9} - 1$ by $- 1$ and simplify.

Tap for more steps...

Step 6.6.1.5.2.1

Divide each term in $- y^{2} > \frac{x^{2}}{9} - 1$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- y^{2}}{- 1} < \frac{\frac{x^{2}}{9}}{- 1} + \frac{- 1}{- 1}$

Step 6.6.1.5.2.2

Simplify the left side.

Tap for more steps...

Step 6.6.1.5.2.2.1

Dividing two negative values results in a positive value.

$\frac{y^{2}}{1} < \frac{\frac{x^{2}}{9}}{- 1} + \frac{- 1}{- 1}$

Step 6.6.1.5.2.2.2

Divide $y^{2}$ by $1$ .

$y^{2} < \frac{\frac{x^{2}}{9}}{- 1} + \frac{- 1}{- 1}$

Step 6.6.1.5.2.3

Simplify the right side.

Tap for more steps...

Step 6.6.1.5.2.3.1

Simplify each term.

Tap for more steps...

Step 6.6.1.5.2.3.1.1

Move the negative one from the denominator of $\frac{\frac{x^{2}}{9}}{- 1}$ .

$y^{2} < - 1 \cdot \frac{x^{2}}{9} + \frac{- 1}{- 1}$

Step 6.6.1.5.2.3.1.2

Rewrite $- 1 \cdot \frac{x^{2}}{9}$ as $- \frac{x^{2}}{9}$ .

$y^{2} < - \frac{x^{2}}{9} + \frac{- 1}{- 1}$

Step 6.6.1.5.2.3.1.3

Divide $- 1$ by $- 1$ .

$y^{2} < - \frac{x^{2}}{9} + 1$

Step 6.6.1.5.3

Take the specified root of both sides of the inequality to eliminate the exponent on the left side.

$\sqrt{y^{2}} < \sqrt{- \frac{x^{2}}{9} + 1}$

Step 6.6.1.5.4

Simplify the equation.

Tap for more steps...

Step 6.6.1.5.4.1

Simplify the left side.

Tap for more steps...

Step 6.6.1.5.4.1.1

Pull terms out from under the radical.

$| y | < \sqrt{- \frac{x^{2}}{9} + 1}$

Step 6.6.1.5.4.2

Simplify the right side.

Tap for more steps...

Step 6.6.1.5.4.2.1

Simplify $\sqrt{- \frac{x^{2}}{9} + 1}$ .

Tap for more steps...

Step 6.6.1.5.4.2.1.1

Simplify the expression.

Tap for more steps...

Step 6.6.1.5.4.2.1.1.1

Rewrite $1$ as $1^{2}$ .

$| y | < \sqrt{- \frac{x^{2}}{9} + 1^{2}}$

Step 6.6.1.5.4.2.1.1.2

Rewrite $\frac{x^{2}}{9}$ as ${(\frac{x}{3})}^{2}$ .

$| y | < \sqrt{- {(\frac{x}{3})}^{2} + 1^{2}}$

Step 6.6.1.5.4.2.1.1.3

Reorder $- {(\frac{x}{3})}^{2}$ and $1^{2}$ .

$| y | < \sqrt{1^{2} - {(\frac{x}{3})}^{2}}$

Step 6.6.1.5.4.2.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = \frac{x}{3}$ .

$| y | < \sqrt{(1 + \frac{x}{3}) (1 - \frac{x}{3})}$

Step 6.6.1.5.4.2.1.3

Simplify terms.

Tap for more steps...

Step 6.6.1.5.4.2.1.3.1

Write $1$ as a fraction with a common denominator.

$| y | < \sqrt{(\frac{3}{3} + \frac{x}{3}) (1 - \frac{x}{3})}$

Step 6.6.1.5.4.2.1.3.2

Combine the numerators over the common denominator.

$| y | < \sqrt{\frac{3 + x}{3} (1 - \frac{x}{3})}$

Step 6.6.1.5.4.2.1.3.3

Write $1$ as a fraction with a common denominator.

$| y | < \sqrt{\frac{3 + x}{3} (\frac{3}{3} - \frac{x}{3})}$

Step 6.6.1.5.4.2.1.3.4

Combine the numerators over the common denominator.

$| y | < \sqrt{\frac{3 + x}{3} \cdot \frac{3 - x}{3}}$

Step 6.6.1.5.4.2.1.3.5

Multiply $\frac{3 + x}{3}$ by $\frac{3 - x}{3}$ .

$| y | < \sqrt{\frac{(3 + x) (3 - x)}{3 \cdot 3}}$

Step 6.6.1.5.4.2.1.3.6

Multiply $3$ by $3$ .

$| y | < \sqrt{\frac{(3 + x) (3 - x)}{9}}$

Step 6.6.1.5.4.2.1.4

Rewrite $\frac{(3 + x) (3 - x)}{9}$ as ${(\frac{1}{3})}^{2} ((3 + x) (3 - x))$ .

Tap for more steps...

Step 6.6.1.5.4.2.1.4.1

Factor the perfect power $1^{2}$ out of $(3 + x) (3 - x)$ .

$| y | < \sqrt{\frac{1^{2} ((3 + x) (3 - x))}{9}}$

Step 6.6.1.5.4.2.1.4.2

Factor the perfect power $3^{2}$ out of $9$ .

$| y | < \sqrt{\frac{1^{2} ((3 + x) (3 - x))}{3^{2} \cdot 1}}$

Step 6.6.1.5.4.2.1.4.3

Rearrange the fraction $\frac{1^{2} ((3 + x) (3 - x))}{3^{2} \cdot 1}$ .

$| y | < \sqrt{{(\frac{1}{3})}^{2} ((3 + x) (3 - x))}$

Step 6.6.1.5.4.2.1.5

Pull terms out from under the radical.

$| y | < | \frac{1}{3} | \sqrt{(3 + x) (3 - x)}$

Step 6.6.1.5.4.2.1.6

$\frac{1}{3}$ is approximately $0.\overline{3}$ which is positive so remove the absolute value

$| y | < \frac{1}{3} \sqrt{(3 + x) (3 - x)}$

Step 6.6.1.5.4.2.1.7

Combine $\frac{1}{3}$ and $\sqrt{(3 + x) (3 - x)}$ .

$| y | < \frac{\sqrt{(3 + x) (3 - x)}}{3}$

Step 6.6.1.5.5

Write $| y | < \frac{\sqrt{(3 + x) (3 - x)}}{3}$ as a piecewise.

Tap for more steps...

Step 6.6.1.5.5.1

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$y \geq 0$

Step 6.6.1.5.5.2

In the piece where $y$ is non-negative, remove the absolute value.

$y < \frac{\sqrt{(3 + x) (3 - x)}}{3}$

Step 6.6.1.5.5.3

Find the domain of $y < \frac{\sqrt{(3 + x) (3 - x)}}{3}$ and find the intersection with $y \geq 0$ .

Tap for more steps...

Step 6.6.1.5.5.3.1

Find the domain of $y < \frac{\sqrt{(3 + x) (3 - x)}}{3}$ .

Tap for more steps...

Step 6.6.1.5.5.3.1.1

Set the radicand in $\sqrt{(3 + x) (3 - x)}$ greater than or equal to $0$ to find where the expression is defined.

$(3 + x) (3 - x) \geq 0$

Step 6.6.1.5.5.3.1.2

Solve for $y$ .

Tap for more steps...

Step 6.6.1.5.5.3.1.2.1

Simplify $(3 + x) (3 - x)$ .

Tap for more steps...

Step 6.6.1.5.5.3.1.2.1.1

Expand $(3 + x) (3 - x)$ using the FOIL Method.

Tap for more steps...

Step 6.6.1.5.5.3.1.2.1.1.1

Apply the distributive property.

$3 (3 - x) + x (3 - x) \geq 0$

Step 6.6.1.5.5.3.1.2.1.1.2

Apply the distributive property.

$3 \cdot 3 + 3 (- x) + x (3 - x) \geq 0$

Step 6.6.1.5.5.3.1.2.1.1.3

Apply the distributive property.

$3 \cdot 3 + 3 (- x) + x \cdot 3 + x (- x) \geq 0$

Step 6.6.1.5.5.3.1.2.1.2

Simplify and combine like terms.

Tap for more steps...

Step 6.6.1.5.5.3.1.2.1.2.1

Simplify each term.

Tap for more steps...

Step 6.6.1.5.5.3.1.2.1.2.1.1

Multiply $3$ by $3$ .

$9 + 3 (- x) + x \cdot 3 + x (- x) \geq 0$

Step 6.6.1.5.5.3.1.2.1.2.1.2

Multiply $- 1$ by $3$ .

$9 - 3 x + x \cdot 3 + x (- x) \geq 0$

Step 6.6.1.5.5.3.1.2.1.2.1.3

Move $3$ to the left of $x$ .

$9 - 3 x + 3 \cdot x + x (- x) \geq 0$

Step 6.6.1.5.5.3.1.2.1.2.1.4

Rewrite using the commutative property of multiplication.

$9 - 3 x + 3 x - x \cdot x \geq 0$

Step 6.6.1.5.5.3.1.2.1.2.1.5

Multiply $x$ by $x$ by adding the exponents.

Tap for more steps...

Step 6.6.1.5.5.3.1.2.1.2.1.5.1

Move $x$ .

$9 - 3 x + 3 x - (x \cdot x) \geq 0$

Step 6.6.1.5.5.3.1.2.1.2.1.5.2

Multiply $x$ by $x$ .

$9 - 3 x + 3 x - x^{2} \geq 0$

Step 6.6.1.5.5.3.1.2.1.2.2

Add $- 3 x$ and $3 x$ .

$9 + 0 - x^{2} \geq 0$

Step 6.6.1.5.5.3.1.2.1.2.3

Add $9$ and $0$ .

$9 - x^{2} \geq 0$

Step 6.6.1.5.5.3.1.2.2

Subtract $9$ from both sides of the inequality.

$- x^{2} \geq - 9$

Step 6.6.1.5.5.3.1.2.3

Divide each term in $- x^{2} \geq - 9$ by $- 1$ and simplify.

Tap for more steps...

Step 6.6.1.5.5.3.1.2.3.1

Divide each term in $- x^{2} \geq - 9$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x^{2}}{- 1} \leq \frac{- 9}{- 1}$

Step 6.6.1.5.5.3.1.2.3.2

Simplify the left side.

Tap for more steps...

Step 6.6.1.5.5.3.1.2.3.2.1

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} \leq \frac{- 9}{- 1}$

Step 6.6.1.5.5.3.1.2.3.2.2

Divide $x^{2}$ by $1$ .

$x^{2} \leq \frac{- 9}{- 1}$

Step 6.6.1.5.5.3.1.2.3.3

Simplify the right side.

Tap for more steps...

Step 6.6.1.5.5.3.1.2.3.3.1

Divide $- 9$ by $- 1$ .

$x^{2} \leq 9$

Step 6.6.1.5.5.3.1.2.4

Take the specified root of both sides of the inequality to eliminate the exponent on the left side.

$\sqrt{x^{2}} \leq \sqrt{9}$

Step 6.6.1.5.5.3.1.2.5

Simplify the equation.

Tap for more steps...

Step 6.6.1.5.5.3.1.2.5.1

Simplify the left side.

Tap for more steps...

Step 6.6.1.5.5.3.1.2.5.1.1

Pull terms out from under the radical.

$| x | \leq \sqrt{9}$

Step 6.6.1.5.5.3.1.2.5.2

Simplify the right side.

Tap for more steps...

Step 6.6.1.5.5.3.1.2.5.2.1

Simplify $\sqrt{9}$ .

Tap for more steps...

Step 6.6.1.5.5.3.1.2.5.2.1.1

Rewrite $9$ as $3^{2}$ .

$| x | \leq \sqrt{3^{2}}$

Step 6.6.1.5.5.3.1.2.5.2.1.2

Pull terms out from under the radical.

$| x | \leq | 3 |$

Step 6.6.1.5.5.3.1.2.5.2.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

$| x | \leq 3$

Step 6.6.1.5.5.3.1.2.6

Write $| x | \leq 3$ as a piecewise.

Tap for more steps...

Step 6.6.1.5.5.3.1.2.6.1

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$x \geq 0$

Step 6.6.1.5.5.3.1.2.6.2

In the piece where $x$ is non-negative, remove the absolute value.

$x \leq 3$

Step 6.6.1.5.5.3.1.2.6.3

To find the interval for the second piece, find where the inside of the absolute value is negative.

$x < 0$

Step 6.6.1.5.5.3.1.2.6.4

In the piece where $x$ is negative, remove the absolute value and multiply by $- 1$ .

$- x \leq 3$

Step 6.6.1.5.5.3.1.2.6.5

Write as a piecewise.

${\begin{cases} x \leq 3 & x \geq 0 \\ - x \leq 3 & x < 0 \end{cases}$

Step 6.6.1.5.5.3.1.2.7

Find the intersection of $x \leq 3$ and $x \geq 0$ .

$0 \leq x \leq 3$

Step 6.6.1.5.5.3.1.2.8

Solve $- x \leq 3$ when $x < 0$ .

Tap for more steps...

Step 6.6.1.5.5.3.1.2.8.1

Divide each term in $- x \leq 3$ by $- 1$ and simplify.

Tap for more steps...

Step 6.6.1.5.5.3.1.2.8.1.1

Divide each term in $- x \leq 3$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \geq \frac{3}{- 1}$

Step 6.6.1.5.5.3.1.2.8.1.2

Simplify the left side.

Tap for more steps...

Step 6.6.1.5.5.3.1.2.8.1.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} \geq \frac{3}{- 1}$

Step 6.6.1.5.5.3.1.2.8.1.2.2

Divide $x$ by $1$ .

$x \geq \frac{3}{- 1}$

Step 6.6.1.5.5.3.1.2.8.1.3

Simplify the right side.

Tap for more steps...

Step 6.6.1.5.5.3.1.2.8.1.3.1

Divide $3$ by $- 1$ .

$x \geq - 3$

Step 6.6.1.5.5.3.1.2.8.2

Find the intersection of $x \geq - 3$ and $x < 0$ .

$- 3 \leq x < 0$

Step 6.6.1.5.5.3.1.2.9

Find the union of the solutions.

$- 3 \leq x \leq 3$

Step 6.6.1.5.5.3.1.3

The domain is all values of $y$ that make the expression defined.

$[- 3, 3]$

Step 6.6.1.5.5.3.2

Find the intersection of $y \geq 0$ and $[- 3, 3]$ .

$0 \leq y \leq 3$

Step 6.6.1.5.5.4

To find the interval for the second piece, find where the inside of the absolute value is negative.

$y < 0$

Step 6.6.1.5.5.5

In the piece where $y$ is negative, remove the absolute value and multiply by $- 1$ .

$- y < \frac{\sqrt{(3 + x) (3 - x)}}{3}$

Step 6.6.1.5.5.6

Find the domain of $- y < \frac{\sqrt{(3 + x) (3 - x)}}{3}$ and find the intersection with $y < 0$ .

Tap for more steps...

Step 6.6.1.5.5.6.1

Find the domain of $- y < \frac{\sqrt{(3 + x) (3 - x)}}{3}$ .

Tap for more steps...

Step 6.6.1.5.5.6.1.1

Set the radicand in $\sqrt{(3 + x) (3 - x)}$ greater than or equal to $0$ to find where the expression is defined.

$(3 + x) (3 - x) \geq 0$

Step 6.6.1.5.5.6.1.2

Solve for $y$ .

Tap for more steps...

Step 6.6.1.5.5.6.1.2.1

Simplify $(3 + x) (3 - x)$ .

Tap for more steps...

Step 6.6.1.5.5.6.1.2.1.1

Expand $(3 + x) (3 - x)$ using the FOIL Method.

Tap for more steps...

Step 6.6.1.5.5.6.1.2.1.1.1

Apply the distributive property.

$3 (3 - x) + x (3 - x) \geq 0$

Step 6.6.1.5.5.6.1.2.1.1.2

Apply the distributive property.

$3 \cdot 3 + 3 (- x) + x (3 - x) \geq 0$

Step 6.6.1.5.5.6.1.2.1.1.3

Apply the distributive property.

$3 \cdot 3 + 3 (- x) + x \cdot 3 + x (- x) \geq 0$

Step 6.6.1.5.5.6.1.2.1.2

Simplify and combine like terms.

Tap for more steps...

Step 6.6.1.5.5.6.1.2.1.2.1

Simplify each term.

Tap for more steps...

Step 6.6.1.5.5.6.1.2.1.2.1.1

Multiply $3$ by $3$ .

$9 + 3 (- x) + x \cdot 3 + x (- x) \geq 0$

Step 6.6.1.5.5.6.1.2.1.2.1.2

Multiply $- 1$ by $3$ .

$9 - 3 x + x \cdot 3 + x (- x) \geq 0$

Step 6.6.1.5.5.6.1.2.1.2.1.3

Move $3$ to the left of $x$ .

$9 - 3 x + 3 \cdot x + x (- x) \geq 0$

Step 6.6.1.5.5.6.1.2.1.2.1.4

Rewrite using the commutative property of multiplication.

$9 - 3 x + 3 x - x \cdot x \geq 0$

Step 6.6.1.5.5.6.1.2.1.2.1.5

Multiply $x$ by $x$ by adding the exponents.

Tap for more steps...

Step 6.6.1.5.5.6.1.2.1.2.1.5.1

Move $x$ .

$9 - 3 x + 3 x - (x \cdot x) \geq 0$

Step 6.6.1.5.5.6.1.2.1.2.1.5.2

Multiply $x$ by $x$ .

$9 - 3 x + 3 x - x^{2} \geq 0$

Step 6.6.1.5.5.6.1.2.1.2.2

Add $- 3 x$ and $3 x$ .

$9 + 0 - x^{2} \geq 0$

Step 6.6.1.5.5.6.1.2.1.2.3

Add $9$ and $0$ .

$9 - x^{2} \geq 0$

Step 6.6.1.5.5.6.1.2.2

Subtract $9$ from both sides of the inequality.

$- x^{2} \geq - 9$

Step 6.6.1.5.5.6.1.2.3

Divide each term in $- x^{2} \geq - 9$ by $- 1$ and simplify.

Tap for more steps...

Step 6.6.1.5.5.6.1.2.3.1

Divide each term in $- x^{2} \geq - 9$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x^{2}}{- 1} \leq \frac{- 9}{- 1}$

Step 6.6.1.5.5.6.1.2.3.2

Simplify the left side.

Tap for more steps...

Step 6.6.1.5.5.6.1.2.3.2.1

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} \leq \frac{- 9}{- 1}$

Step 6.6.1.5.5.6.1.2.3.2.2

Divide $x^{2}$ by $1$ .

$x^{2} \leq \frac{- 9}{- 1}$

Step 6.6.1.5.5.6.1.2.3.3

Simplify the right side.

Tap for more steps...

Step 6.6.1.5.5.6.1.2.3.3.1

Divide $- 9$ by $- 1$ .

$x^{2} \leq 9$

Step 6.6.1.5.5.6.1.2.4

Take the specified root of both sides of the inequality to eliminate the exponent on the left side.

$\sqrt{x^{2}} \leq \sqrt{9}$

Step 6.6.1.5.5.6.1.2.5

Simplify the equation.

Tap for more steps...

Step 6.6.1.5.5.6.1.2.5.1

Simplify the left side.

Tap for more steps...

Step 6.6.1.5.5.6.1.2.5.1.1

Pull terms out from under the radical.

$| x | \leq \sqrt{9}$

Step 6.6.1.5.5.6.1.2.5.2

Simplify the right side.

Tap for more steps...

Step 6.6.1.5.5.6.1.2.5.2.1

Simplify $\sqrt{9}$ .

Tap for more steps...

Step 6.6.1.5.5.6.1.2.5.2.1.1

Rewrite $9$ as $3^{2}$ .

$| x | \leq \sqrt{3^{2}}$

Step 6.6.1.5.5.6.1.2.5.2.1.2

Pull terms out from under the radical.

$| x | \leq | 3 |$

Step 6.6.1.5.5.6.1.2.5.2.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

$| x | \leq 3$

Step 6.6.1.5.5.6.1.2.6

Write $| x | \leq 3$ as a piecewise.

Tap for more steps...

Step 6.6.1.5.5.6.1.2.6.1

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$x \geq 0$

Step 6.6.1.5.5.6.1.2.6.2

In the piece where $x$ is non-negative, remove the absolute value.

$x \leq 3$

Step 6.6.1.5.5.6.1.2.6.3

To find the interval for the second piece, find where the inside of the absolute value is negative.

$x < 0$

Step 6.6.1.5.5.6.1.2.6.4

In the piece where $x$ is negative, remove the absolute value and multiply by $- 1$ .

$- x \leq 3$

Step 6.6.1.5.5.6.1.2.6.5

Write as a piecewise.

${\begin{cases} x \leq 3 & x \geq 0 \\ - x \leq 3 & x < 0 \end{cases}$

Step 6.6.1.5.5.6.1.2.7

Find the intersection of $x \leq 3$ and $x \geq 0$ .

$0 \leq x \leq 3$

Step 6.6.1.5.5.6.1.2.8

Solve $- x \leq 3$ when $x < 0$ .

Tap for more steps...

Step 6.6.1.5.5.6.1.2.8.1

Divide each term in $- x \leq 3$ by $- 1$ and simplify.

Tap for more steps...

Step 6.6.1.5.5.6.1.2.8.1.1

Divide each term in $- x \leq 3$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \geq \frac{3}{- 1}$

Step 6.6.1.5.5.6.1.2.8.1.2

Simplify the left side.

Tap for more steps...

Step 6.6.1.5.5.6.1.2.8.1.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} \geq \frac{3}{- 1}$

Step 6.6.1.5.5.6.1.2.8.1.2.2

Divide $x$ by $1$ .

$x \geq \frac{3}{- 1}$

Step 6.6.1.5.5.6.1.2.8.1.3

Simplify the right side.

Tap for more steps...

Step 6.6.1.5.5.6.1.2.8.1.3.1

Divide $3$ by $- 1$ .

$x \geq - 3$

Step 6.6.1.5.5.6.1.2.8.2

Find the intersection of $x \geq - 3$ and $x < 0$ .

$- 3 \leq x < 0$

Step 6.6.1.5.5.6.1.2.9

Find the union of the solutions.

$- 3 \leq x \leq 3$

Step 6.6.1.5.5.6.1.3

The domain is all values of $y$ that make the expression defined.

$[- 3, 3]$

Step 6.6.1.5.5.6.2

Find the intersection of $y < 0$ and $[- 3, 3]$ .

$- 3 \leq y < 0$

Step 6.6.1.5.5.7

Write as a piecewise.

${\begin{cases} y < \frac{\sqrt{(3 + x) (3 - x)}}{3} & 0 \leq y \leq 3 \\ - y < \frac{\sqrt{(3 + x) (3 - x)}}{3} & - 3 \leq y < 0 \end{cases}$

Step 6.6.1.5.6

Find the intersection of $y < \frac{\sqrt{(3 + x) (3 - x)}}{3}$ and $0 \leq y \leq 3$ .

$0 \leq y \leq N o (Min i m u m)$

Step 6.6.1.5.7

Solve $- y < \frac{\sqrt{(3 + x) (3 - x)}}{3}$ when $- 3 \leq y < 0$ .

Tap for more steps...

Step 6.6.1.5.7.1

Divide each term in $- y < \frac{\sqrt{(3 + x) (3 - x)}}{3}$ by $- 1$ and simplify.

Tap for more steps...

Step 6.6.1.5.7.1.1

Divide each term in $- y < \frac{\sqrt{(3 + x) (3 - x)}}{3}$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- y}{- 1} > \frac{\frac{\sqrt{(3 + x) (3 - x)}}{3}}{- 1}$

Step 6.6.1.5.7.1.2

Simplify the left side.

Tap for more steps...

Step 6.6.1.5.7.1.2.1

Dividing two negative values results in a positive value.

$\frac{y}{1} > \frac{\frac{\sqrt{(3 + x) (3 - x)}}{3}}{- 1}$

Step 6.6.1.5.7.1.2.2

Divide $y$ by $1$ .

$y > \frac{\frac{\sqrt{(3 + x) (3 - x)}}{3}}{- 1}$

Step 6.6.1.5.7.1.3

Simplify the right side.

Tap for more steps...

Step 6.6.1.5.7.1.3.1

Move the negative one from the denominator of $\frac{\frac{\sqrt{(3 + x) (3 - x)}}{3}}{- 1}$ .

$y > - 1 \cdot \frac{\sqrt{(3 + x) (3 - x)}}{3}$

Step 6.6.1.5.7.1.3.2

Rewrite $- 1 \cdot \frac{\sqrt{(3 + x) (3 - x)}}{3}$ as $- \frac{\sqrt{(3 + x) (3 - x)}}{3}$ .

$y > - \frac{\sqrt{(3 + x) (3 - x)}}{3}$

Step 6.6.1.5.7.2

Find the intersection of $y > - \frac{\sqrt{(3 + x) (3 - x)}}{3}$ and $- 3 \leq y < 0$ .

No solution

Step 6.6.1.5.8

Find the union of the solutions.

$0 \leq y \leq i N o Min m^{2} u$

Step 6.6.2

Find the intersection of $0 \leq y \leq i N o Min m^{2} u$ and $0 \leq x \leq 1$ .

$0 \leq x \leq N o (Min i m u m)$

Step 6.7

Solve $- x < 3 \sqrt{(1 + y) (1 - y)}$ when $- 1 \leq x < 0$ .

Tap for more steps...

Step 6.7.1

Solve $- x < 3 \sqrt{(1 + y) (1 - y)}$ for $y$ .

Tap for more steps...

Step 6.7.1.1

Rewrite so $y$ is on the left side of the inequality.

$3 \sqrt{(1 + y) (1 - y)} > - x$

Step 6.7.1.2

Divide each term in $3 \sqrt{(1 + y) (1 - y)} > - x$ by $3$ and simplify.

Tap for more steps...

Step 6.7.1.2.1

Divide each term in $3 \sqrt{(1 + y) (1 - y)} > - x$ by $3$ .

$\frac{3 \sqrt{(1 + y) (1 - y)}}{3} > \frac{- x}{3}$

Step 6.7.1.2.2

Simplify the left side.

Tap for more steps...

Step 6.7.1.2.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 6.7.1.2.2.1.1

Cancel the common factor.

$\frac{3 \sqrt{(1 + y) (1 - y)}}{3} > \frac{- x}{3}$

Step 6.7.1.2.2.1.2

Divide $\sqrt{(1 + y) (1 - y)}$ by $1$ .

$\sqrt{(1 + y) (1 - y)} > \frac{- x}{3}$

Step 6.7.1.2.3

Simplify the right side.

Tap for more steps...

Step 6.7.1.2.3.1

Move the negative in front of the fraction.

$\sqrt{(1 + y) (1 - y)} > - \frac{x}{3}$

Step 6.7.1.3

To remove the radical on the left side of the inequality, square both sides of the inequality.

${\sqrt{(1 + y) (1 - y)}}^{2} > {(- \frac{x}{3})}^{2}$

Step 6.7.1.4

Simplify each side of the inequality.

Tap for more steps...

Step 6.7.1.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{(1 + y) (1 - y)}$ as ${((1 + y) (1 - y))}^{\frac{1}{2}}$ .

${({((1 + y) (1 - y))}^{\frac{1}{2}})}^{2} > {(- \frac{x}{3})}^{2}$

Step 6.7.1.4.2

Simplify the left side.

Tap for more steps...

Step 6.7.1.4.2.1

Simplify ${({((1 + y) (1 - y))}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 6.7.1.4.2.1.1

Multiply the exponents in ${({((1 + y) (1 - y))}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 6.7.1.4.2.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${((1 + y) (1 - y))}^{\frac{1}{2} \cdot 2} > {(- \frac{x}{3})}^{2}$

Step 6.7.1.4.2.1.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 6.7.1.4.2.1.1.2.1

Cancel the common factor.

${((1 + y) (1 - y))}^{\frac{1}{2} \cdot 2} > {(- \frac{x}{3})}^{2}$

Step 6.7.1.4.2.1.1.2.2

Rewrite the expression.

${((1 + y) (1 - y))}^{1} > {(- \frac{x}{3})}^{2}$

Step 6.7.1.4.2.1.2

Expand $(1 + y) (1 - y)$ using the FOIL Method.

Tap for more steps...

Step 6.7.1.4.2.1.2.1

Apply the distributive property.

${(1 (1 - y) + y (1 - y))}^{1} > {(- \frac{x}{3})}^{2}$

Step 6.7.1.4.2.1.2.2

Apply the distributive property.

${(1 \cdot 1 + 1 (- y) + y (1 - y))}^{1} > {(- \frac{x}{3})}^{2}$

Step 6.7.1.4.2.1.2.3

Apply the distributive property.

${(1 \cdot 1 + 1 (- y) + y \cdot 1 + y (- y))}^{1} > {(- \frac{x}{3})}^{2}$

Step 6.7.1.4.2.1.3

Simplify and combine like terms.

Tap for more steps...

Step 6.7.1.4.2.1.3.1

Simplify each term.

Tap for more steps...

Step 6.7.1.4.2.1.3.1.1

Multiply $1$ by $1$ .

${(1 + 1 (- y) + y \cdot 1 + y (- y))}^{1} > {(- \frac{x}{3})}^{2}$

Step 6.7.1.4.2.1.3.1.2

Multiply $- y$ by $1$ .

${(1 - y + y \cdot 1 + y (- y))}^{1} > {(- \frac{x}{3})}^{2}$

Step 6.7.1.4.2.1.3.1.3

Multiply $y$ by $1$ .

${(1 - y + y + y (- y))}^{1} > {(- \frac{x}{3})}^{2}$

Step 6.7.1.4.2.1.3.1.4

Rewrite using the commutative property of multiplication.

${(1 - y + y - y \cdot y)}^{1} > {(- \frac{x}{3})}^{2}$

Step 6.7.1.4.2.1.3.1.5

Multiply $y$ by $y$ by adding the exponents.

Tap for more steps...

Step 6.7.1.4.2.1.3.1.5.1

Move $y$ .

${(1 - y + y - (y \cdot y))}^{1} > {(- \frac{x}{3})}^{2}$

Step 6.7.1.4.2.1.3.1.5.2

Multiply $y$ by $y$ .

${(1 - y + y - y^{2})}^{1} > {(- \frac{x}{3})}^{2}$

Step 6.7.1.4.2.1.3.2

Add $- y$ and $y$ .

${(1 + 0 - y^{2})}^{1} > {(- \frac{x}{3})}^{2}$

Step 6.7.1.4.2.1.3.3

Add $1$ and $0$ .

${(1 - y^{2})}^{1} > {(- \frac{x}{3})}^{2}$

Step 6.7.1.4.2.1.4

Simplify.

$1 - y^{2} > {(- \frac{x}{3})}^{2}$

Step 6.7.1.4.3

Simplify the right side.

Tap for more steps...

Step 6.7.1.4.3.1

Simplify ${(- \frac{x}{3})}^{2}$ .

Tap for more steps...

Step 6.7.1.4.3.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Tap for more steps...

Step 6.7.1.4.3.1.1.1

Apply the product rule to $- \frac{x}{3}$ .

$1 - y^{2} > {(- 1)}^{2} {(\frac{x}{3})}^{2}$

Step 6.7.1.4.3.1.1.2

Apply the product rule to $\frac{x}{3}$ .

$1 - y^{2} > {(- 1)}^{2} \frac{x^{2}}{3^{2}}$

Step 6.7.1.4.3.1.2

Raise $- 1$ to the power of $2$ .

$1 - y^{2} > 1 \frac{x^{2}}{3^{2}}$

Step 6.7.1.4.3.1.3

Multiply $\frac{x^{2}}{3^{2}}$ by $1$ .

$1 - y^{2} > \frac{x^{2}}{3^{2}}$

Step 6.7.1.4.3.1.4

Raise $3$ to the power of $2$ .

$1 - y^{2} > \frac{x^{2}}{9}$

Step 6.7.1.5

Solve for $y$ .

Tap for more steps...

Step 6.7.1.5.1

Subtract $1$ from both sides of the inequality.

$- y^{2} > \frac{x^{2}}{9} - 1$

Step 6.7.1.5.2

Divide each term in $- y^{2} > \frac{x^{2}}{9} - 1$ by $- 1$ and simplify.

Tap for more steps...

Step 6.7.1.5.2.1

Divide each term in $- y^{2} > \frac{x^{2}}{9} - 1$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- y^{2}}{- 1} < \frac{\frac{x^{2}}{9}}{- 1} + \frac{- 1}{- 1}$

Step 6.7.1.5.2.2

Simplify the left side.

Tap for more steps...

Step 6.7.1.5.2.2.1

Dividing two negative values results in a positive value.

$\frac{y^{2}}{1} < \frac{\frac{x^{2}}{9}}{- 1} + \frac{- 1}{- 1}$

Step 6.7.1.5.2.2.2

Divide $y^{2}$ by $1$ .

$y^{2} < \frac{\frac{x^{2}}{9}}{- 1} + \frac{- 1}{- 1}$

Step 6.7.1.5.2.3

Simplify the right side.

Tap for more steps...

Step 6.7.1.5.2.3.1

Simplify each term.

Tap for more steps...

Step 6.7.1.5.2.3.1.1

Move the negative one from the denominator of $\frac{\frac{x^{2}}{9}}{- 1}$ .

$y^{2} < - 1 \cdot \frac{x^{2}}{9} + \frac{- 1}{- 1}$

Step 6.7.1.5.2.3.1.2

Rewrite $- 1 \cdot \frac{x^{2}}{9}$ as $- \frac{x^{2}}{9}$ .

$y^{2} < - \frac{x^{2}}{9} + \frac{- 1}{- 1}$

Step 6.7.1.5.2.3.1.3

Divide $- 1$ by $- 1$ .

$y^{2} < - \frac{x^{2}}{9} + 1$

Step 6.7.1.5.3

Take the specified root of both sides of the inequality to eliminate the exponent on the left side.

$\sqrt{y^{2}} < \sqrt{- \frac{x^{2}}{9} + 1}$

Step 6.7.1.5.4

Simplify the equation.

Tap for more steps...

Step 6.7.1.5.4.1

Simplify the left side.

Tap for more steps...

Step 6.7.1.5.4.1.1

Pull terms out from under the radical.

$| y | < \sqrt{- \frac{x^{2}}{9} + 1}$

Step 6.7.1.5.4.2

Simplify the right side.

Tap for more steps...

Step 6.7.1.5.4.2.1

Simplify $\sqrt{- \frac{x^{2}}{9} + 1}$ .

Tap for more steps...

Step 6.7.1.5.4.2.1.1

Simplify the expression.

Tap for more steps...

Step 6.7.1.5.4.2.1.1.1

Rewrite $1$ as $1^{2}$ .

$| y | < \sqrt{- \frac{x^{2}}{9} + 1^{2}}$

Step 6.7.1.5.4.2.1.1.2

Rewrite $\frac{x^{2}}{9}$ as ${(\frac{x}{3})}^{2}$ .

$| y | < \sqrt{- {(\frac{x}{3})}^{2} + 1^{2}}$

Step 6.7.1.5.4.2.1.1.3

Reorder $- {(\frac{x}{3})}^{2}$ and $1^{2}$ .

$| y | < \sqrt{1^{2} - {(\frac{x}{3})}^{2}}$

Step 6.7.1.5.4.2.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = \frac{x}{3}$ .

$| y | < \sqrt{(1 + \frac{x}{3}) (1 - \frac{x}{3})}$

Step 6.7.1.5.4.2.1.3

Simplify terms.

Tap for more steps...

Step 6.7.1.5.4.2.1.3.1

Write $1$ as a fraction with a common denominator.

$| y | < \sqrt{(\frac{3}{3} + \frac{x}{3}) (1 - \frac{x}{3})}$

Step 6.7.1.5.4.2.1.3.2

Combine the numerators over the common denominator.

$| y | < \sqrt{\frac{3 + x}{3} (1 - \frac{x}{3})}$

Step 6.7.1.5.4.2.1.3.3

Write $1$ as a fraction with a common denominator.

$| y | < \sqrt{\frac{3 + x}{3} (\frac{3}{3} - \frac{x}{3})}$

Step 6.7.1.5.4.2.1.3.4

Combine the numerators over the common denominator.

$| y | < \sqrt{\frac{3 + x}{3} \cdot \frac{3 - x}{3}}$

Step 6.7.1.5.4.2.1.3.5

Multiply $\frac{3 + x}{3}$ by $\frac{3 - x}{3}$ .

$| y | < \sqrt{\frac{(3 + x) (3 - x)}{3 \cdot 3}}$

Step 6.7.1.5.4.2.1.3.6

Multiply $3$ by $3$ .

$| y | < \sqrt{\frac{(3 + x) (3 - x)}{9}}$

Step 6.7.1.5.4.2.1.4

Rewrite $\frac{(3 + x) (3 - x)}{9}$ as ${(\frac{1}{3})}^{2} ((3 + x) (3 - x))$ .

Tap for more steps...

Step 6.7.1.5.4.2.1.4.1

Factor the perfect power $1^{2}$ out of $(3 + x) (3 - x)$ .

$| y | < \sqrt{\frac{1^{2} ((3 + x) (3 - x))}{9}}$

Step 6.7.1.5.4.2.1.4.2

Factor the perfect power $3^{2}$ out of $9$ .

$| y | < \sqrt{\frac{1^{2} ((3 + x) (3 - x))}{3^{2} \cdot 1}}$

Step 6.7.1.5.4.2.1.4.3

Rearrange the fraction $\frac{1^{2} ((3 + x) (3 - x))}{3^{2} \cdot 1}$ .

$| y | < \sqrt{{(\frac{1}{3})}^{2} ((3 + x) (3 - x))}$

Step 6.7.1.5.4.2.1.5

Pull terms out from under the radical.

$| y | < | \frac{1}{3} | \sqrt{(3 + x) (3 - x)}$

Step 6.7.1.5.4.2.1.6

$\frac{1}{3}$ is approximately $0.\overline{3}$ which is positive so remove the absolute value

$| y | < \frac{1}{3} \sqrt{(3 + x) (3 - x)}$

Step 6.7.1.5.4.2.1.7

Combine $\frac{1}{3}$ and $\sqrt{(3 + x) (3 - x)}$ .

$| y | < \frac{\sqrt{(3 + x) (3 - x)}}{3}$

Step 6.7.1.5.5

Write $| y | < \frac{\sqrt{(3 + x) (3 - x)}}{3}$ as a piecewise.

Tap for more steps...

Step 6.7.1.5.5.1

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$y \geq 0$

Step 6.7.1.5.5.2

In the piece where $y$ is non-negative, remove the absolute value.

$y < \frac{\sqrt{(3 + x) (3 - x)}}{3}$

Step 6.7.1.5.5.3

Find the domain of $y < \frac{\sqrt{(3 + x) (3 - x)}}{3}$ and find the intersection with $y \geq 0$ .

Tap for more steps...

Step 6.7.1.5.5.3.1

Find the domain of $y < \frac{\sqrt{(3 + x) (3 - x)}}{3}$ .

Tap for more steps...

Step 6.7.1.5.5.3.1.1

Set the radicand in $\sqrt{(3 + x) (3 - x)}$ greater than or equal to $0$ to find where the expression is defined.

$(3 + x) (3 - x) \geq 0$

Step 6.7.1.5.5.3.1.2

Solve for $y$ .

Tap for more steps...

Step 6.7.1.5.5.3.1.2.1

Simplify $(3 + x) (3 - x)$ .

Tap for more steps...

Step 6.7.1.5.5.3.1.2.1.1

Expand $(3 + x) (3 - x)$ using the FOIL Method.

Tap for more steps...

Step 6.7.1.5.5.3.1.2.1.1.1

Apply the distributive property.

$3 (3 - x) + x (3 - x) \geq 0$

Step 6.7.1.5.5.3.1.2.1.1.2

Apply the distributive property.

$3 \cdot 3 + 3 (- x) + x (3 - x) \geq 0$

Step 6.7.1.5.5.3.1.2.1.1.3

Apply the distributive property.

$3 \cdot 3 + 3 (- x) + x \cdot 3 + x (- x) \geq 0$

Step 6.7.1.5.5.3.1.2.1.2

Simplify and combine like terms.

Tap for more steps...

Step 6.7.1.5.5.3.1.2.1.2.1

Simplify each term.

Tap for more steps...

Step 6.7.1.5.5.3.1.2.1.2.1.1

Multiply $3$ by $3$ .

$9 + 3 (- x) + x \cdot 3 + x (- x) \geq 0$

Step 6.7.1.5.5.3.1.2.1.2.1.2

Multiply $- 1$ by $3$ .

$9 - 3 x + x \cdot 3 + x (- x) \geq 0$

Step 6.7.1.5.5.3.1.2.1.2.1.3

Move $3$ to the left of $x$ .

$9 - 3 x + 3 \cdot x + x (- x) \geq 0$

Step 6.7.1.5.5.3.1.2.1.2.1.4

Rewrite using the commutative property of multiplication.

$9 - 3 x + 3 x - x \cdot x \geq 0$

Step 6.7.1.5.5.3.1.2.1.2.1.5

Multiply $x$ by $x$ by adding the exponents.

Tap for more steps...

Step 6.7.1.5.5.3.1.2.1.2.1.5.1

Move $x$ .

$9 - 3 x + 3 x - (x \cdot x) \geq 0$

Step 6.7.1.5.5.3.1.2.1.2.1.5.2

Multiply $x$ by $x$ .

$9 - 3 x + 3 x - x^{2} \geq 0$

Step 6.7.1.5.5.3.1.2.1.2.2

Add $- 3 x$ and $3 x$ .

$9 + 0 - x^{2} \geq 0$

Step 6.7.1.5.5.3.1.2.1.2.3

Add $9$ and $0$ .

$9 - x^{2} \geq 0$

Step 6.7.1.5.5.3.1.2.2

Subtract $9$ from both sides of the inequality.

$- x^{2} \geq - 9$

Step 6.7.1.5.5.3.1.2.3

Divide each term in $- x^{2} \geq - 9$ by $- 1$ and simplify.

Tap for more steps...

Step 6.7.1.5.5.3.1.2.3.1

Divide each term in $- x^{2} \geq - 9$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x^{2}}{- 1} \leq \frac{- 9}{- 1}$

Step 6.7.1.5.5.3.1.2.3.2

Simplify the left side.

Tap for more steps...

Step 6.7.1.5.5.3.1.2.3.2.1

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} \leq \frac{- 9}{- 1}$

Step 6.7.1.5.5.3.1.2.3.2.2

Divide $x^{2}$ by $1$ .

$x^{2} \leq \frac{- 9}{- 1}$

Step 6.7.1.5.5.3.1.2.3.3

Simplify the right side.

Tap for more steps...

Step 6.7.1.5.5.3.1.2.3.3.1

Divide $- 9$ by $- 1$ .

$x^{2} \leq 9$

Step 6.7.1.5.5.3.1.2.4

Take the specified root of both sides of the inequality to eliminate the exponent on the left side.

$\sqrt{x^{2}} \leq \sqrt{9}$

Step 6.7.1.5.5.3.1.2.5

Simplify the equation.

Tap for more steps...

Step 6.7.1.5.5.3.1.2.5.1

Simplify the left side.

Tap for more steps...

Step 6.7.1.5.5.3.1.2.5.1.1

Pull terms out from under the radical.

$| x | \leq \sqrt{9}$

Step 6.7.1.5.5.3.1.2.5.2

Simplify the right side.

Tap for more steps...

Step 6.7.1.5.5.3.1.2.5.2.1

Simplify $\sqrt{9}$ .

Tap for more steps...

Step 6.7.1.5.5.3.1.2.5.2.1.1

Rewrite $9$ as $3^{2}$ .

$| x | \leq \sqrt{3^{2}}$

Step 6.7.1.5.5.3.1.2.5.2.1.2

Pull terms out from under the radical.

$| x | \leq | 3 |$

Step 6.7.1.5.5.3.1.2.5.2.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

$| x | \leq 3$

Step 6.7.1.5.5.3.1.2.6

Write $| x | \leq 3$ as a piecewise.

Tap for more steps...

Step 6.7.1.5.5.3.1.2.6.1

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$x \geq 0$

Step 6.7.1.5.5.3.1.2.6.2

In the piece where $x$ is non-negative, remove the absolute value.

$x \leq 3$

Step 6.7.1.5.5.3.1.2.6.3

To find the interval for the second piece, find where the inside of the absolute value is negative.

$x < 0$

Step 6.7.1.5.5.3.1.2.6.4

In the piece where $x$ is negative, remove the absolute value and multiply by $- 1$ .

$- x \leq 3$

Step 6.7.1.5.5.3.1.2.6.5

Write as a piecewise.

${\begin{cases} x \leq 3 & x \geq 0 \\ - x \leq 3 & x < 0 \end{cases}$

Step 6.7.1.5.5.3.1.2.7

Find the intersection of $x \leq 3$ and $x \geq 0$ .

$0 \leq x \leq 3$

Step 6.7.1.5.5.3.1.2.8

Solve $- x \leq 3$ when $x < 0$ .

Tap for more steps...

Step 6.7.1.5.5.3.1.2.8.1

Divide each term in $- x \leq 3$ by $- 1$ and simplify.

Tap for more steps...

Step 6.7.1.5.5.3.1.2.8.1.1

Divide each term in $- x \leq 3$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \geq \frac{3}{- 1}$

Step 6.7.1.5.5.3.1.2.8.1.2

Simplify the left side.

Tap for more steps...

Step 6.7.1.5.5.3.1.2.8.1.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} \geq \frac{3}{- 1}$

Step 6.7.1.5.5.3.1.2.8.1.2.2

Divide $x$ by $1$ .

$x \geq \frac{3}{- 1}$

Step 6.7.1.5.5.3.1.2.8.1.3

Simplify the right side.

Tap for more steps...

Step 6.7.1.5.5.3.1.2.8.1.3.1

Divide $3$ by $- 1$ .

$x \geq - 3$

Step 6.7.1.5.5.3.1.2.8.2

Find the intersection of $x \geq - 3$ and $x < 0$ .

$- 3 \leq x < 0$

Step 6.7.1.5.5.3.1.2.9

Find the union of the solutions.

$- 3 \leq x \leq 3$

Step 6.7.1.5.5.3.1.3

The domain is all values of $y$ that make the expression defined.

$[- 3, 3]$

Step 6.7.1.5.5.3.2

Find the intersection of $y \geq 0$ and $[- 3, 3]$ .

$0 \leq y \leq 3$

Step 6.7.1.5.5.4

To find the interval for the second piece, find where the inside of the absolute value is negative.

$y < 0$

Step 6.7.1.5.5.5

In the piece where $y$ is negative, remove the absolute value and multiply by $- 1$ .

$- y < \frac{\sqrt{(3 + x) (3 - x)}}{3}$

Step 6.7.1.5.5.6

Find the domain of $- y < \frac{\sqrt{(3 + x) (3 - x)}}{3}$ and find the intersection with $y < 0$ .

Tap for more steps...

Step 6.7.1.5.5.6.1

Find the domain of $- y < \frac{\sqrt{(3 + x) (3 - x)}}{3}$ .

Tap for more steps...

Step 6.7.1.5.5.6.1.1

Set the radicand in $\sqrt{(3 + x) (3 - x)}$ greater than or equal to $0$ to find where the expression is defined.

$(3 + x) (3 - x) \geq 0$

Step 6.7.1.5.5.6.1.2

Solve for $y$ .

Tap for more steps...

Step 6.7.1.5.5.6.1.2.1

Simplify $(3 + x) (3 - x)$ .

Tap for more steps...

Step 6.7.1.5.5.6.1.2.1.1

Expand $(3 + x) (3 - x)$ using the FOIL Method.

Tap for more steps...

Step 6.7.1.5.5.6.1.2.1.1.1

Apply the distributive property.

$3 (3 - x) + x (3 - x) \geq 0$

Step 6.7.1.5.5.6.1.2.1.1.2

Apply the distributive property.

$3 \cdot 3 + 3 (- x) + x (3 - x) \geq 0$

Step 6.7.1.5.5.6.1.2.1.1.3

Apply the distributive property.

$3 \cdot 3 + 3 (- x) + x \cdot 3 + x (- x) \geq 0$

Step 6.7.1.5.5.6.1.2.1.2

Simplify and combine like terms.

Tap for more steps...

Step 6.7.1.5.5.6.1.2.1.2.1

Simplify each term.

Tap for more steps...

Step 6.7.1.5.5.6.1.2.1.2.1.1

Multiply $3$ by $3$ .

$9 + 3 (- x) + x \cdot 3 + x (- x) \geq 0$

Step 6.7.1.5.5.6.1.2.1.2.1.2

Multiply $- 1$ by $3$ .

$9 - 3 x + x \cdot 3 + x (- x) \geq 0$

Step 6.7.1.5.5.6.1.2.1.2.1.3

Move $3$ to the left of $x$ .

$9 - 3 x + 3 \cdot x + x (- x) \geq 0$

Step 6.7.1.5.5.6.1.2.1.2.1.4

Rewrite using the commutative property of multiplication.

$9 - 3 x + 3 x - x \cdot x \geq 0$

Step 6.7.1.5.5.6.1.2.1.2.1.5

Multiply $x$ by $x$ by adding the exponents.

Tap for more steps...

Step 6.7.1.5.5.6.1.2.1.2.1.5.1

Move $x$ .

$9 - 3 x + 3 x - (x \cdot x) \geq 0$

Step 6.7.1.5.5.6.1.2.1.2.1.5.2

Multiply $x$ by $x$ .

$9 - 3 x + 3 x - x^{2} \geq 0$

Step 6.7.1.5.5.6.1.2.1.2.2

Add $- 3 x$ and $3 x$ .

$9 + 0 - x^{2} \geq 0$

Step 6.7.1.5.5.6.1.2.1.2.3

Add $9$ and $0$ .

$9 - x^{2} \geq 0$

Step 6.7.1.5.5.6.1.2.2

Subtract $9$ from both sides of the inequality.

$- x^{2} \geq - 9$

Step 6.7.1.5.5.6.1.2.3

Divide each term in $- x^{2} \geq - 9$ by $- 1$ and simplify.

Tap for more steps...

Step 6.7.1.5.5.6.1.2.3.1

Divide each term in $- x^{2} \geq - 9$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x^{2}}{- 1} \leq \frac{- 9}{- 1}$

Step 6.7.1.5.5.6.1.2.3.2

Simplify the left side.

Tap for more steps...

Step 6.7.1.5.5.6.1.2.3.2.1

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} \leq \frac{- 9}{- 1}$

Step 6.7.1.5.5.6.1.2.3.2.2

Divide $x^{2}$ by $1$ .

$x^{2} \leq \frac{- 9}{- 1}$

Step 6.7.1.5.5.6.1.2.3.3

Simplify the right side.

Tap for more steps...

Step 6.7.1.5.5.6.1.2.3.3.1

Divide $- 9$ by $- 1$ .

$x^{2} \leq 9$

Step 6.7.1.5.5.6.1.2.4

Take the specified root of both sides of the inequality to eliminate the exponent on the left side.

$\sqrt{x^{2}} \leq \sqrt{9}$

Step 6.7.1.5.5.6.1.2.5

Simplify the equation.

Tap for more steps...

Step 6.7.1.5.5.6.1.2.5.1

Simplify the left side.

Tap for more steps...

Step 6.7.1.5.5.6.1.2.5.1.1

Pull terms out from under the radical.

$| x | \leq \sqrt{9}$

Step 6.7.1.5.5.6.1.2.5.2

Simplify the right side.

Tap for more steps...

Step 6.7.1.5.5.6.1.2.5.2.1

Simplify $\sqrt{9}$ .

Tap for more steps...

Step 6.7.1.5.5.6.1.2.5.2.1.1

Rewrite $9$ as $3^{2}$ .

$| x | \leq \sqrt{3^{2}}$

Step 6.7.1.5.5.6.1.2.5.2.1.2

Pull terms out from under the radical.

$| x | \leq | 3 |$

Step 6.7.1.5.5.6.1.2.5.2.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

$| x | \leq 3$

Step 6.7.1.5.5.6.1.2.6

Write $| x | \leq 3$ as a piecewise.

Tap for more steps...

Step 6.7.1.5.5.6.1.2.6.1

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$x \geq 0$

Step 6.7.1.5.5.6.1.2.6.2

In the piece where $x$ is non-negative, remove the absolute value.

$x \leq 3$

Step 6.7.1.5.5.6.1.2.6.3

To find the interval for the second piece, find where the inside of the absolute value is negative.

$x < 0$

Step 6.7.1.5.5.6.1.2.6.4

In the piece where $x$ is negative, remove the absolute value and multiply by $- 1$ .

$- x \leq 3$

Step 6.7.1.5.5.6.1.2.6.5

Write as a piecewise.

${\begin{cases} x \leq 3 & x \geq 0 \\ - x \leq 3 & x < 0 \end{cases}$

Step 6.7.1.5.5.6.1.2.7

Find the intersection of $x \leq 3$ and $x \geq 0$ .

$0 \leq x \leq 3$

Step 6.7.1.5.5.6.1.2.8

Solve $- x \leq 3$ when $x < 0$ .

Tap for more steps...

Step 6.7.1.5.5.6.1.2.8.1

Divide each term in $- x \leq 3$ by $- 1$ and simplify.

Tap for more steps...

Step 6.7.1.5.5.6.1.2.8.1.1

Divide each term in $- x \leq 3$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \geq \frac{3}{- 1}$

Step 6.7.1.5.5.6.1.2.8.1.2

Simplify the left side.

Tap for more steps...

Step 6.7.1.5.5.6.1.2.8.1.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} \geq \frac{3}{- 1}$

Step 6.7.1.5.5.6.1.2.8.1.2.2

Divide $x$ by $1$ .

$x \geq \frac{3}{- 1}$

Step 6.7.1.5.5.6.1.2.8.1.3

Simplify the right side.

Tap for more steps...

Step 6.7.1.5.5.6.1.2.8.1.3.1

Divide $3$ by $- 1$ .

$x \geq - 3$

Step 6.7.1.5.5.6.1.2.8.2

Find the intersection of $x \geq - 3$ and $x < 0$ .

$- 3 \leq x < 0$

Step 6.7.1.5.5.6.1.2.9

Find the union of the solutions.

$- 3 \leq x \leq 3$

Step 6.7.1.5.5.6.1.3

The domain is all values of $y$ that make the expression defined.

$[- 3, 3]$

Step 6.7.1.5.5.6.2

Find the intersection of $y < 0$ and $[- 3, 3]$ .

$- 3 \leq y < 0$

Step 6.7.1.5.5.7

Write as a piecewise.

${\begin{cases} y < \frac{\sqrt{(3 + x) (3 - x)}}{3} & 0 \leq y \leq 3 \\ - y < \frac{\sqrt{(3 + x) (3 - x)}}{3} & - 3 \leq y < 0 \end{cases}$

Step 6.7.1.5.6

Find the intersection of $y < \frac{\sqrt{(3 + x) (3 - x)}}{3}$ and $0 \leq y \leq 3$ .

$0 \leq y \leq N o (Min i m u m)$

Step 6.7.1.5.7

Solve $- y < \frac{\sqrt{(3 + x) (3 - x)}}{3}$ when $- 3 \leq y < 0$ .

Tap for more steps...

Step 6.7.1.5.7.1

Divide each term in $- y < \frac{\sqrt{(3 + x) (3 - x)}}{3}$ by $- 1$ and simplify.

Tap for more steps...

Step 6.7.1.5.7.1.1

Divide each term in $- y < \frac{\sqrt{(3 + x) (3 - x)}}{3}$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- y}{- 1} > \frac{\frac{\sqrt{(3 + x) (3 - x)}}{3}}{- 1}$

Step 6.7.1.5.7.1.2

Simplify the left side.

Tap for more steps...

Step 6.7.1.5.7.1.2.1

Dividing two negative values results in a positive value.

$\frac{y}{1} > \frac{\frac{\sqrt{(3 + x) (3 - x)}}{3}}{- 1}$

Step 6.7.1.5.7.1.2.2

Divide $y$ by $1$ .

$y > \frac{\frac{\sqrt{(3 + x) (3 - x)}}{3}}{- 1}$

Step 6.7.1.5.7.1.3

Simplify the right side.

Tap for more steps...

Step 6.7.1.5.7.1.3.1

Move the negative one from the denominator of $\frac{\frac{\sqrt{(3 + x) (3 - x)}}{3}}{- 1}$ .

$y > - 1 \cdot \frac{\sqrt{(3 + x) (3 - x)}}{3}$

Step 6.7.1.5.7.1.3.2

Rewrite $- 1 \cdot \frac{\sqrt{(3 + x) (3 - x)}}{3}$ as $- \frac{\sqrt{(3 + x) (3 - x)}}{3}$ .

$y > - \frac{\sqrt{(3 + x) (3 - x)}}{3}$

Step 6.7.1.5.7.2

Find the intersection of $y > - \frac{\sqrt{(3 + x) (3 - x)}}{3}$ and $- 3 \leq y < 0$ .

No solution

Step 6.7.1.5.8

Find the union of the solutions.

$0 \leq y \leq i N o Min m^{2} u$

Step 6.7.2

Find the intersection of $0 \leq y \leq i N o Min m^{2} u$ and $- 1 \leq x < 0$ .

$0 \leq x < N o (Min i m u m)$

Step 6.8

Find the union of the solutions.

$0 \leq x < N o (Max i m u m)$

Step 7

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression defined.

No solution