Pre-Algebra Examples

$y^{2} + 4 x^{2} > 4$

Step 1

Subtract $4 x^{2}$ from both sides of the inequality.

$y^{2} > 4 - 4 x^{2}$

Step 2

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

$\sqrt{y^{2}} > \sqrt{4 - 4 x^{2}}$

Step 3

Simplify the equation.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Pull terms out from under the radical.

$| y | > \sqrt{4 - 4 x^{2}}$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $\sqrt{4 - 4 x^{2}}$ .

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Step 3.2.1.1

Factor $4$ out of $4 - 4 x^{2}$ .

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Step 3.2.1.1.1

Factor $4$ out of $4$ .

$| y | > \sqrt{4 (1) - 4 x^{2}}$

Step 3.2.1.1.2

Factor $4$ out of $- 4 x^{2}$ .

$| y | > \sqrt{4 (1) + 4 (- x^{2})}$

Step 3.2.1.1.3

Factor $4$ out of $4 (1) + 4 (- x^{2})$ .

$| y | > \sqrt{4 (1 - x^{2})}$

Step 3.2.1.2

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

$| y | > \sqrt{4 (1^{2} - x^{2})}$

Step 3.2.1.3

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 $b = x$ .

$| y | > \sqrt{4 (1 + x) (1 - x)}$

Step 3.2.1.4

Rewrite $4 (1 + x) (1 - x)$ as $2^{2} ((1^{2} + x) (1 - x))$ .

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Step 3.2.1.4.1

Rewrite $4$ as $2^{2}$ .

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

Step 3.2.1.4.2

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

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

Step 3.2.1.4.3

Add parentheses.

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

Step 3.2.1.5

Pull terms out from under the radical.

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

Step 3.2.1.6

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

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

Step 3.2.1.7

One to any power is one.

$| y | > 2 \sqrt{(1 + x) (1 - x)}$

Step 4

Write $| y | > 2 \sqrt{(1 + x) (1 - x)}$ as a piecewise.

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Step 4.1

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

$y \geq 0$

Step 4.2

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

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

Step 4.3

Find the domain of $y > 2 \sqrt{(1 + x) (1 - x)}$ and find the intersection with $y \geq 0$ .

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Step 4.3.1

Find the domain of $y > 2 \sqrt{(1 + x) (1 - x)}$ .

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Step 4.3.1.1

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

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

Step 4.3.1.2

Solve for $y$ .

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Step 4.3.1.2.1

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

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Step 4.3.1.2.1.1

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

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Step 4.3.1.2.1.1.1

Apply the distributive property.

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

Step 4.3.1.2.1.1.2

Apply the distributive property.

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

Step 4.3.1.2.1.1.3

Apply the distributive property.

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

Step 4.3.1.2.1.2

Simplify and combine like terms.

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Step 4.3.1.2.1.2.1

Simplify each term.

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Step 4.3.1.2.1.2.1.1

Multiply $1$ by $1$ .

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

Step 4.3.1.2.1.2.1.2

Multiply $- x$ by $1$ .

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

Step 4.3.1.2.1.2.1.3

Multiply $x$ by $1$ .

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

Step 4.3.1.2.1.2.1.4

Rewrite using the commutative property of multiplication.

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

Step 4.3.1.2.1.2.1.5

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

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Step 4.3.1.2.1.2.1.5.1

Move $x$ .

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

Step 4.3.1.2.1.2.1.5.2

Multiply $x$ by $x$ .

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

Step 4.3.1.2.1.2.2

Add $- x$ and $x$ .

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

Step 4.3.1.2.1.2.3

Add $1$ and $0$ .

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

Step 4.3.1.2.2

Subtract $1$ from both sides of the inequality.

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

Step 4.3.1.2.3

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

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Step 4.3.1.2.3.1

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

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

Step 4.3.1.2.3.2

Simplify the left side.

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Step 4.3.1.2.3.2.1

Dividing two negative values results in a positive value.

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

Step 4.3.1.2.3.2.2

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

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

Step 4.3.1.2.3.3

Simplify the right side.

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Step 4.3.1.2.3.3.1

Divide $- 1$ by $- 1$ .

$x^{2} \leq 1$

Step 4.3.1.2.4

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

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

Step 4.3.1.2.5

Simplify the equation.

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Step 4.3.1.2.5.1

Simplify the left side.

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Step 4.3.1.2.5.1.1

Pull terms out from under the radical.

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

Step 4.3.1.2.5.2

Simplify the right side.

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Step 4.3.1.2.5.2.1

Any root of $1$ is $1$ .

$| x | \leq 1$

Step 4.3.1.2.6

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

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Step 4.3.1.2.6.1

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

$x \geq 0$

Step 4.3.1.2.6.2

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

$x \leq 1$

Step 4.3.1.2.6.3

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

$x < 0$

Step 4.3.1.2.6.4

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

$- x \leq 1$

Step 4.3.1.2.6.5

Write as a piecewise.

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

Step 4.3.1.2.7

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

$0 \leq x \leq 1$

Step 4.3.1.2.8

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

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Step 4.3.1.2.8.1

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

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Step 4.3.1.2.8.1.1

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

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

Step 4.3.1.2.8.1.2

Simplify the left side.

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Step 4.3.1.2.8.1.2.1

Dividing two negative values results in a positive value.

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

Step 4.3.1.2.8.1.2.2

Divide $x$ by $1$ .

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

Step 4.3.1.2.8.1.3

Simplify the right side.

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Step 4.3.1.2.8.1.3.1

Divide $1$ by $- 1$ .

$x \geq - 1$

Step 4.3.1.2.8.2

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

$- 1 \leq x < 0$

Step 4.3.1.2.9

Find the union of the solutions.

$- 1 \leq x \leq 1$

Step 4.3.1.3

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

$[- 1, 1]$

Step 4.3.2

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

$0 \leq y \leq 1$

Step 4.4

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

$y < 0$

Step 4.5

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

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

Step 4.6

Find the domain of $- y > 2 \sqrt{(1 + x) (1 - x)}$ and find the intersection with $y < 0$ .

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Step 4.6.1

Find the domain of $- y > 2 \sqrt{(1 + x) (1 - x)}$ .

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Step 4.6.1.1

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

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

Step 4.6.1.2

Solve for $y$ .

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Step 4.6.1.2.1

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

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Step 4.6.1.2.1.1

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

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Step 4.6.1.2.1.1.1

Apply the distributive property.

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

Step 4.6.1.2.1.1.2

Apply the distributive property.

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

Step 4.6.1.2.1.1.3

Apply the distributive property.

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

Step 4.6.1.2.1.2

Simplify and combine like terms.

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Step 4.6.1.2.1.2.1

Simplify each term.

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Step 4.6.1.2.1.2.1.1

Multiply $1$ by $1$ .

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

Step 4.6.1.2.1.2.1.2

Multiply $- x$ by $1$ .

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

Step 4.6.1.2.1.2.1.3

Multiply $x$ by $1$ .

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

Step 4.6.1.2.1.2.1.4

Rewrite using the commutative property of multiplication.

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

Step 4.6.1.2.1.2.1.5

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

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Step 4.6.1.2.1.2.1.5.1

Move $x$ .

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

Step 4.6.1.2.1.2.1.5.2

Multiply $x$ by $x$ .

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

Step 4.6.1.2.1.2.2

Add $- x$ and $x$ .

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

Step 4.6.1.2.1.2.3

Add $1$ and $0$ .

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

Step 4.6.1.2.2

Subtract $1$ from both sides of the inequality.

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

Step 4.6.1.2.3

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

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Step 4.6.1.2.3.1

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

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

Step 4.6.1.2.3.2

Simplify the left side.

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Step 4.6.1.2.3.2.1

Dividing two negative values results in a positive value.

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

Step 4.6.1.2.3.2.2

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

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

Step 4.6.1.2.3.3

Simplify the right side.

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Step 4.6.1.2.3.3.1

Divide $- 1$ by $- 1$ .

$x^{2} \leq 1$

Step 4.6.1.2.4

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

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

Step 4.6.1.2.5

Simplify the equation.

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Step 4.6.1.2.5.1

Simplify the left side.

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Step 4.6.1.2.5.1.1

Pull terms out from under the radical.

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

Step 4.6.1.2.5.2

Simplify the right side.

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Step 4.6.1.2.5.2.1

Any root of $1$ is $1$ .

$| x | \leq 1$

Step 4.6.1.2.6

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

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Step 4.6.1.2.6.1

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

$x \geq 0$

Step 4.6.1.2.6.2

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

$x \leq 1$

Step 4.6.1.2.6.3

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

$x < 0$

Step 4.6.1.2.6.4

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

$- x \leq 1$

Step 4.6.1.2.6.5

Write as a piecewise.

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

Step 4.6.1.2.7

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

$0 \leq x \leq 1$

Step 4.6.1.2.8

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

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Step 4.6.1.2.8.1

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

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Step 4.6.1.2.8.1.1

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

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

Step 4.6.1.2.8.1.2

Simplify the left side.

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Step 4.6.1.2.8.1.2.1

Dividing two negative values results in a positive value.

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

Step 4.6.1.2.8.1.2.2

Divide $x$ by $1$ .

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

Step 4.6.1.2.8.1.3

Simplify the right side.

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Step 4.6.1.2.8.1.3.1

Divide $1$ by $- 1$ .

$x \geq - 1$

Step 4.6.1.2.8.2

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

$- 1 \leq x < 0$

Step 4.6.1.2.9

Find the union of the solutions.

$- 1 \leq x \leq 1$

Step 4.6.1.3

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

$[- 1, 1]$

Step 4.6.2

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

$- 1 \leq y < 0$

Step 4.7

Write as a piecewise.

${\begin{cases} y > 2 \sqrt{(1 + x) (1 - x)} & 0 \leq y \leq 1 \\ - y > 2 \sqrt{(1 + x) (1 - x)} & - 1 \leq y < 0 \end{cases}$

Step 5

Find the intersection of $y > 2 \sqrt{(1 + x) (1 - x)}$ and $0 \leq y \leq 1$ .

No solution

Step 6

Solve $- y > 2 \sqrt{(1 + x) (1 - x)}$ when $- 1 \leq y < 0$ .

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Step 6.1

Divide each term in $- y > 2 \sqrt{(1 + x) (1 - x)}$ by $- 1$ and simplify.

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Step 6.1.1

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

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

Step 6.1.2

Simplify the left side.

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Step 6.1.2.1

Dividing two negative values results in a positive value.

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

Step 6.1.2.2

Divide $y$ by $1$ .

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

Step 6.1.3

Simplify the right side.

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Step 6.1.3.1

Move the negative one from the denominator of $\frac{2 \sqrt{(1 + x) (1 - x)}}{- 1}$ .

$y < - 1 \cdot (2 \sqrt{(1 + x) (1 - x)})$

Step 6.1.3.2

Rewrite $- 1 \cdot (2 \sqrt{(1 + x) (1 - x)})$ as $- (2 \sqrt{(1 + x) (1 - x)})$ .

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

Step 6.1.3.3

Multiply $2$ by $- 1$ .

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

Step 6.2

Find the intersection of $y < - 2 \sqrt{(1 + x) (1 - x)}$ and $- 1 \leq y < 0$ .

$- 1 \leq y < N o (Min i m u m)$

Step 7

Find the union of the solutions.

$- 1 \leq y < i N o Min m^{2} u$

Step 8