Finite Math Examples

$x^{2} + y^{2} > 49$

Step 1

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

$x^{2} > 49 - y^{2}$

Step 2

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

$\sqrt{x^{2}} > \sqrt{49 - y^{2}}$

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.

$| x | > \sqrt{49 - y^{2}}$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $\sqrt{49 - y^{2}}$ .

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Step 3.2.1.1

Rewrite $49$ as $7^{2}$ .

$| x | > \sqrt{7^{2} - y^{2}}$

Step 3.2.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 7$ and $b = y$ .

$| x | > \sqrt{(7 + y) (7 - y)}$

Step 4

Write $| x | > \sqrt{(7 + y) (7 - y)}$ 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.

$x \geq 0$

Step 4.2

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

$x > \sqrt{(7 + y) (7 - y)}$

Step 4.3

Find the domain of $x > \sqrt{(7 + y) (7 - y)}$ and find the intersection with $x \geq 0$ .

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Step 4.3.1

Find the domain of $x > \sqrt{(7 + y) (7 - y)}$ .

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Step 4.3.1.1

Set the radicand in $\sqrt{(7 + y) (7 - y)}$ greater than or equal to $0$ to find where the expression is defined.

$(7 + y) (7 - y) \geq 0$

Step 4.3.1.2

Solve for $x$ .

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Step 4.3.1.2.1

Simplify $(7 + y) (7 - y)$ .

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Step 4.3.1.2.1.1

Expand $(7 + y) (7 - y)$ using the FOIL Method.

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Step 4.3.1.2.1.1.1

Apply the distributive property.

$7 (7 - y) + y (7 - y) \geq 0$

Step 4.3.1.2.1.1.2

Apply the distributive property.

$7 \cdot 7 + 7 (- y) + y (7 - y) \geq 0$

Step 4.3.1.2.1.1.3

Apply the distributive property.

$7 \cdot 7 + 7 (- y) + y \cdot 7 + y (- y) \geq 0$

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 $7$ by $7$ .

$49 + 7 (- y) + y \cdot 7 + y (- y) \geq 0$

Step 4.3.1.2.1.2.1.2

Multiply $- 1$ by $7$ .

$49 - 7 y + y \cdot 7 + y (- y) \geq 0$

Step 4.3.1.2.1.2.1.3

Move $7$ to the left of $y$ .

$49 - 7 y + 7 \cdot y + y (- y) \geq 0$

Step 4.3.1.2.1.2.1.4

Rewrite using the commutative property of multiplication.

$49 - 7 y + 7 y - y \cdot y \geq 0$

Step 4.3.1.2.1.2.1.5

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

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Step 4.3.1.2.1.2.1.5.1

Move $y$ .

$49 - 7 y + 7 y - (y \cdot y) \geq 0$

Step 4.3.1.2.1.2.1.5.2

Multiply $y$ by $y$ .

$49 - 7 y + 7 y - y^{2} \geq 0$

Step 4.3.1.2.1.2.2

Add $- 7 y$ and $7 y$ .

$49 + 0 - y^{2} \geq 0$

Step 4.3.1.2.1.2.3

Add $49$ and $0$ .

$49 - y^{2} \geq 0$

Step 4.3.1.2.2

Subtract $49$ from both sides of the inequality.

$- y^{2} \geq - 49$

Step 4.3.1.2.3

Divide each term in $- y^{2} \geq - 49$ by $- 1$ and simplify.

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Step 4.3.1.2.3.1

Divide each term in $- y^{2} \geq - 49$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- y^{2}}{- 1} \leq \frac{- 49}{- 1}$

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.

$\frac{y^{2}}{1} \leq \frac{- 49}{- 1}$

Step 4.3.1.2.3.2.2

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

$y^{2} \leq \frac{- 49}{- 1}$

Step 4.3.1.2.3.3

Simplify the right side.

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Step 4.3.1.2.3.3.1

Divide $- 49$ by $- 1$ .

$y^{2} \leq 49$

Step 4.3.1.2.4

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

$\sqrt{y^{2}} \leq \sqrt{49}$

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.

$| y | \leq \sqrt{49}$

Step 4.3.1.2.5.2

Simplify the right side.

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Step 4.3.1.2.5.2.1

Simplify $\sqrt{49}$ .

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Step 4.3.1.2.5.2.1.1

Rewrite $49$ as $7^{2}$ .

$| y | \leq \sqrt{7^{2}}$

Step 4.3.1.2.5.2.1.2

Pull terms out from under the radical.

$| y | \leq | 7 |$

Step 4.3.1.2.5.2.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $7$ is $7$ .

$| y | \leq 7$

Step 4.3.1.2.6

Write $| y | \leq 7$ 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.

$y \geq 0$

Step 4.3.1.2.6.2

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

$y \leq 7$

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.

$y < 0$

Step 4.3.1.2.6.4

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

$- y \leq 7$

Step 4.3.1.2.6.5

Write as a piecewise.

${\begin{cases} y \leq 7 & y \geq 0 \\ - y \leq 7 & y < 0 \end{cases}$

Step 4.3.1.2.7

Find the intersection of $y \leq 7$ and $y \geq 0$ .

$0 \leq y \leq 7$

Step 4.3.1.2.8

Solve $- y \leq 7$ when $y < 0$ .

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Step 4.3.1.2.8.1

Divide each term in $- y \leq 7$ by $- 1$ and simplify.

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Step 4.3.1.2.8.1.1

Divide each term in $- y \leq 7$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- y}{- 1} \geq \frac{7}{- 1}$

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.

$\frac{y}{1} \geq \frac{7}{- 1}$

Step 4.3.1.2.8.1.2.2

Divide $y$ by $1$ .

$y \geq \frac{7}{- 1}$

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 $7$ by $- 1$ .

$y \geq - 7$

Step 4.3.1.2.8.2

Find the intersection of $y \geq - 7$ and $y < 0$ .

$- 7 \leq y < 0$

Step 4.3.1.2.9

Find the union of the solutions.

$- 7 \leq y \leq 7$

Step 4.3.1.3

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

$[- 7, 7]$

Step 4.3.2

Find the intersection of $x \geq 0$ and $[- 7, 7]$ .

$0 \leq x \leq 7$

Step 4.4

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

$x < 0$

Step 4.5

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

$- x > \sqrt{(7 + y) (7 - y)}$

Step 4.6

Find the domain of $- x > \sqrt{(7 + y) (7 - y)}$ and find the intersection with $x < 0$ .

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Step 4.6.1

Find the domain of $- x > \sqrt{(7 + y) (7 - y)}$ .

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Step 4.6.1.1

Set the radicand in $\sqrt{(7 + y) (7 - y)}$ greater than or equal to $0$ to find where the expression is defined.

$(7 + y) (7 - y) \geq 0$

Step 4.6.1.2

Solve for $x$ .

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Step 4.6.1.2.1

Simplify $(7 + y) (7 - y)$ .

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Step 4.6.1.2.1.1

Expand $(7 + y) (7 - y)$ using the FOIL Method.

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Step 4.6.1.2.1.1.1

Apply the distributive property.

$7 (7 - y) + y (7 - y) \geq 0$

Step 4.6.1.2.1.1.2

Apply the distributive property.

$7 \cdot 7 + 7 (- y) + y (7 - y) \geq 0$

Step 4.6.1.2.1.1.3

Apply the distributive property.

$7 \cdot 7 + 7 (- y) + y \cdot 7 + y (- y) \geq 0$

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 $7$ by $7$ .

$49 + 7 (- y) + y \cdot 7 + y (- y) \geq 0$

Step 4.6.1.2.1.2.1.2

Multiply $- 1$ by $7$ .

$49 - 7 y + y \cdot 7 + y (- y) \geq 0$

Step 4.6.1.2.1.2.1.3

Move $7$ to the left of $y$ .

$49 - 7 y + 7 \cdot y + y (- y) \geq 0$

Step 4.6.1.2.1.2.1.4

Rewrite using the commutative property of multiplication.

$49 - 7 y + 7 y - y \cdot y \geq 0$

Step 4.6.1.2.1.2.1.5

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

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Step 4.6.1.2.1.2.1.5.1

Move $y$ .

$49 - 7 y + 7 y - (y \cdot y) \geq 0$

Step 4.6.1.2.1.2.1.5.2

Multiply $y$ by $y$ .

$49 - 7 y + 7 y - y^{2} \geq 0$

Step 4.6.1.2.1.2.2

Add $- 7 y$ and $7 y$ .

$49 + 0 - y^{2} \geq 0$

Step 4.6.1.2.1.2.3

Add $49$ and $0$ .

$49 - y^{2} \geq 0$

Step 4.6.1.2.2

Subtract $49$ from both sides of the inequality.

$- y^{2} \geq - 49$

Step 4.6.1.2.3

Divide each term in $- y^{2} \geq - 49$ by $- 1$ and simplify.

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Step 4.6.1.2.3.1

Divide each term in $- y^{2} \geq - 49$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- y^{2}}{- 1} \leq \frac{- 49}{- 1}$

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.

$\frac{y^{2}}{1} \leq \frac{- 49}{- 1}$

Step 4.6.1.2.3.2.2

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

$y^{2} \leq \frac{- 49}{- 1}$

Step 4.6.1.2.3.3

Simplify the right side.

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Step 4.6.1.2.3.3.1

Divide $- 49$ by $- 1$ .

$y^{2} \leq 49$

Step 4.6.1.2.4

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

$\sqrt{y^{2}} \leq \sqrt{49}$

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.

$| y | \leq \sqrt{49}$

Step 4.6.1.2.5.2

Simplify the right side.

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Step 4.6.1.2.5.2.1

Simplify $\sqrt{49}$ .

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Step 4.6.1.2.5.2.1.1

Rewrite $49$ as $7^{2}$ .

$| y | \leq \sqrt{7^{2}}$

Step 4.6.1.2.5.2.1.2

Pull terms out from under the radical.

$| y | \leq | 7 |$

Step 4.6.1.2.5.2.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $7$ is $7$ .

$| y | \leq 7$

Step 4.6.1.2.6

Write $| y | \leq 7$ 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.

$y \geq 0$

Step 4.6.1.2.6.2

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

$y \leq 7$

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.

$y < 0$

Step 4.6.1.2.6.4

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

$- y \leq 7$

Step 4.6.1.2.6.5

Write as a piecewise.

${\begin{cases} y \leq 7 & y \geq 0 \\ - y \leq 7 & y < 0 \end{cases}$

Step 4.6.1.2.7

Find the intersection of $y \leq 7$ and $y \geq 0$ .

$0 \leq y \leq 7$

Step 4.6.1.2.8

Solve $- y \leq 7$ when $y < 0$ .

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Step 4.6.1.2.8.1

Divide each term in $- y \leq 7$ by $- 1$ and simplify.

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Step 4.6.1.2.8.1.1

Divide each term in $- y \leq 7$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- y}{- 1} \geq \frac{7}{- 1}$

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.

$\frac{y}{1} \geq \frac{7}{- 1}$

Step 4.6.1.2.8.1.2.2

Divide $y$ by $1$ .

$y \geq \frac{7}{- 1}$

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 $7$ by $- 1$ .

$y \geq - 7$

Step 4.6.1.2.8.2

Find the intersection of $y \geq - 7$ and $y < 0$ .

$- 7 \leq y < 0$

Step 4.6.1.2.9

Find the union of the solutions.

$- 7 \leq y \leq 7$

Step 4.6.1.3

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

$[- 7, 7]$

Step 4.6.2

Find the intersection of $x < 0$ and $[- 7, 7]$ .

$- 7 \leq x < 0$

Step 4.7

Write as a piecewise.

${\begin{cases} x > \sqrt{(7 + y) (7 - y)} & 0 \leq x \leq 7 \\ - x > \sqrt{(7 + y) (7 - y)} & - 7 \leq x < 0 \end{cases}$

Step 5

Solve $x > \sqrt{(7 + y) (7 - y)}$ when $0 \leq x \leq 7$ .

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Step 5.1

Solve $x > \sqrt{(7 + y) (7 - y)}$ for $y$ .

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Step 5.1.1

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

$\sqrt{(7 + y) (7 - y)} < x$

Step 5.1.2

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

${\sqrt{(7 + y) (7 - y)}}^{2} < x^{2}$

Step 5.1.3

Simplify each side of the inequality.

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Step 5.1.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{(7 + y) (7 - y)}$ as ${((7 + y) (7 - y))}^{\frac{1}{2}}$ .

${({((7 + y) (7 - y))}^{\frac{1}{2}})}^{2} < x^{2}$

Step 5.1.3.2

Simplify the left side.

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Step 5.1.3.2.1

Simplify ${({((7 + y) (7 - y))}^{\frac{1}{2}})}^{2}$ .

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Step 5.1.3.2.1.1

Multiply the exponents in ${({((7 + y) (7 - y))}^{\frac{1}{2}})}^{2}$ .

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Step 5.1.3.2.1.1.1

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

${((7 + y) (7 - y))}^{\frac{1}{2} \cdot 2} < x^{2}$

Step 5.1.3.2.1.1.2

Cancel the common factor of $2$ .

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Step 5.1.3.2.1.1.2.1

Cancel the common factor.

${((7 + y) (7 - y))}^{\frac{1}{2} \cdot 2} < x^{2}$

Step 5.1.3.2.1.1.2.2

Rewrite the expression.

${((7 + y) (7 - y))}^{1} < x^{2}$

Step 5.1.3.2.1.2

Expand $(7 + y) (7 - y)$ using the FOIL Method.

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Step 5.1.3.2.1.2.1

Apply the distributive property.

${(7 (7 - y) + y (7 - y))}^{1} < x^{2}$

Step 5.1.3.2.1.2.2

Apply the distributive property.

${(7 \cdot 7 + 7 (- y) + y (7 - y))}^{1} < x^{2}$

Step 5.1.3.2.1.2.3

Apply the distributive property.

${(7 \cdot 7 + 7 (- y) + y \cdot 7 + y (- y))}^{1} < x^{2}$

Step 5.1.3.2.1.3

Simplify and combine like terms.

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Step 5.1.3.2.1.3.1

Simplify each term.

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Step 5.1.3.2.1.3.1.1

Multiply $7$ by $7$ .

${(49 + 7 (- y) + y \cdot 7 + y (- y))}^{1} < x^{2}$

Step 5.1.3.2.1.3.1.2

Multiply $- 1$ by $7$ .

${(49 - 7 y + y \cdot 7 + y (- y))}^{1} < x^{2}$

Step 5.1.3.2.1.3.1.3

Move $7$ to the left of $y$ .

${(49 - 7 y + 7 \cdot y + y (- y))}^{1} < x^{2}$

Step 5.1.3.2.1.3.1.4

Rewrite using the commutative property of multiplication.

${(49 - 7 y + 7 y - y \cdot y)}^{1} < x^{2}$

Step 5.1.3.2.1.3.1.5

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

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Step 5.1.3.2.1.3.1.5.1

Move $y$ .

${(49 - 7 y + 7 y - (y \cdot y))}^{1} < x^{2}$

Step 5.1.3.2.1.3.1.5.2

Multiply $y$ by $y$ .

${(49 - 7 y + 7 y - y^{2})}^{1} < x^{2}$

Step 5.1.3.2.1.3.2

Add $- 7 y$ and $7 y$ .

${(49 + 0 - y^{2})}^{1} < x^{2}$

Step 5.1.3.2.1.3.3

Add $49$ and $0$ .

${(49 - y^{2})}^{1} < x^{2}$

Step 5.1.3.2.1.4

Simplify.

$49 - y^{2} < x^{2}$

Step 5.1.4

Solve for $y$ .

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Step 5.1.4.1

Subtract $49$ from both sides of the inequality.

$- y^{2} < x^{2} - 49$

Step 5.1.4.2

Divide each term in $- y^{2} < x^{2} - 49$ by $- 1$ and simplify.

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Step 5.1.4.2.1

Divide each term in $- y^{2} < x^{2} - 49$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- y^{2}}{- 1} > \frac{x^{2}}{- 1} + \frac{- 49}{- 1}$

Step 5.1.4.2.2

Simplify the left side.

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Step 5.1.4.2.2.1

Dividing two negative values results in a positive value.

$\frac{y^{2}}{1} > \frac{x^{2}}{- 1} + \frac{- 49}{- 1}$

Step 5.1.4.2.2.2

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

$y^{2} > \frac{x^{2}}{- 1} + \frac{- 49}{- 1}$

Step 5.1.4.2.3

Simplify the right side.

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Step 5.1.4.2.3.1

Simplify each term.

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Step 5.1.4.2.3.1.1

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

$y^{2} > - 1 \cdot x^{2} + \frac{- 49}{- 1}$

Step 5.1.4.2.3.1.2

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

$y^{2} > - x^{2} + \frac{- 49}{- 1}$

Step 5.1.4.2.3.1.3

Divide $- 49$ by $- 1$ .

$y^{2} > - x^{2} + 49$

Step 5.1.4.3

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

$\sqrt{y^{2}} > \sqrt{- x^{2} + 49}$

Step 5.1.4.4

Simplify the equation.

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Step 5.1.4.4.1

Simplify the left side.

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Step 5.1.4.4.1.1

Pull terms out from under the radical.

$| y | > \sqrt{- x^{2} + 49}$

Step 5.1.4.4.2

Simplify the right side.

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Step 5.1.4.4.2.1

Simplify $\sqrt{- x^{2} + 49}$ .

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Step 5.1.4.4.2.1.1

Simplify the expression.

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Step 5.1.4.4.2.1.1.1

Rewrite $49$ as $7^{2}$ .

$| y | > \sqrt{- x^{2} + 7^{2}}$

Step 5.1.4.4.2.1.1.2

Reorder $- x^{2}$ and $7^{2}$ .

$| y | > \sqrt{7^{2} - x^{2}}$

Step 5.1.4.4.2.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 7$ and $b = x$ .

$| y | > \sqrt{(7 + x) (7 - x)}$

Step 5.1.4.5

Write $| y | > \sqrt{(7 + x) (7 - x)}$ as a piecewise.

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Step 5.1.4.5.1

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

$y \geq 0$

Step 5.1.4.5.2

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

$y > \sqrt{(7 + x) (7 - x)}$

Step 5.1.4.5.3

Find the domain of $y > \sqrt{(7 + x) (7 - x)}$ and find the intersection with $y \geq 0$ .

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Step 5.1.4.5.3.1

Find the domain of $y > \sqrt{(7 + x) (7 - x)}$ .

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Step 5.1.4.5.3.1.1

Set the radicand in $\sqrt{(7 + x) (7 - x)}$ greater than or equal to $0$ to find where the expression is defined.

$(7 + x) (7 - x) \geq 0$

Step 5.1.4.5.3.1.2

Solve for $y$ .

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Step 5.1.4.5.3.1.2.1

Simplify $(7 + x) (7 - x)$ .

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Step 5.1.4.5.3.1.2.1.1

Expand $(7 + x) (7 - x)$ using the FOIL Method.

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Step 5.1.4.5.3.1.2.1.1.1

Apply the distributive property.

$7 (7 - x) + x (7 - x) \geq 0$

Step 5.1.4.5.3.1.2.1.1.2

Apply the distributive property.

$7 \cdot 7 + 7 (- x) + x (7 - x) \geq 0$

Step 5.1.4.5.3.1.2.1.1.3

Apply the distributive property.

$7 \cdot 7 + 7 (- x) + x \cdot 7 + x (- x) \geq 0$

Step 5.1.4.5.3.1.2.1.2

Simplify and combine like terms.

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Step 5.1.4.5.3.1.2.1.2.1

Simplify each term.

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Step 5.1.4.5.3.1.2.1.2.1.1

Multiply $7$ by $7$ .

$49 + 7 (- x) + x \cdot 7 + x (- x) \geq 0$

Step 5.1.4.5.3.1.2.1.2.1.2

Multiply $- 1$ by $7$ .

$49 - 7 x + x \cdot 7 + x (- x) \geq 0$

Step 5.1.4.5.3.1.2.1.2.1.3

Move $7$ to the left of $x$ .

$49 - 7 x + 7 \cdot x + x (- x) \geq 0$

Step 5.1.4.5.3.1.2.1.2.1.4

Rewrite using the commutative property of multiplication.

$49 - 7 x + 7 x - x \cdot x \geq 0$

Step 5.1.4.5.3.1.2.1.2.1.5

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

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Step 5.1.4.5.3.1.2.1.2.1.5.1

Move $x$ .

$49 - 7 x + 7 x - (x \cdot x) \geq 0$

Step 5.1.4.5.3.1.2.1.2.1.5.2

Multiply $x$ by $x$ .

$49 - 7 x + 7 x - x^{2} \geq 0$

Step 5.1.4.5.3.1.2.1.2.2

Add $- 7 x$ and $7 x$ .

$49 + 0 - x^{2} \geq 0$

Step 5.1.4.5.3.1.2.1.2.3

Add $49$ and $0$ .

$49 - x^{2} \geq 0$

Step 5.1.4.5.3.1.2.2

Subtract $49$ from both sides of the inequality.

$- x^{2} \geq - 49$

Step 5.1.4.5.3.1.2.3

Divide each term in $- x^{2} \geq - 49$ by $- 1$ and simplify.

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Step 5.1.4.5.3.1.2.3.1

Divide each term in $- x^{2} \geq - 49$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x^{2}}{- 1} \leq \frac{- 49}{- 1}$

Step 5.1.4.5.3.1.2.3.2

Simplify the left side.

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Step 5.1.4.5.3.1.2.3.2.1

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} \leq \frac{- 49}{- 1}$

Step 5.1.4.5.3.1.2.3.2.2

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

$x^{2} \leq \frac{- 49}{- 1}$

Step 5.1.4.5.3.1.2.3.3

Simplify the right side.

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Step 5.1.4.5.3.1.2.3.3.1

Divide $- 49$ by $- 1$ .

$x^{2} \leq 49$

Step 5.1.4.5.3.1.2.4

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

$\sqrt{x^{2}} \leq \sqrt{49}$

Step 5.1.4.5.3.1.2.5

Simplify the equation.

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Step 5.1.4.5.3.1.2.5.1

Simplify the left side.

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Step 5.1.4.5.3.1.2.5.1.1

Pull terms out from under the radical.

$| x | \leq \sqrt{49}$

Step 5.1.4.5.3.1.2.5.2

Simplify the right side.

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Step 5.1.4.5.3.1.2.5.2.1

Simplify $\sqrt{49}$ .

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Step 5.1.4.5.3.1.2.5.2.1.1

Rewrite $49$ as $7^{2}$ .

$| x | \leq \sqrt{7^{2}}$

Step 5.1.4.5.3.1.2.5.2.1.2

Pull terms out from under the radical.

$| x | \leq | 7 |$

Step 5.1.4.5.3.1.2.5.2.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $7$ is $7$ .

$| x | \leq 7$

Step 5.1.4.5.3.1.2.6

Write $| x | \leq 7$ as a piecewise.

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Step 5.1.4.5.3.1.2.6.1

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

$x \geq 0$

Step 5.1.4.5.3.1.2.6.2

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

$x \leq 7$

Step 5.1.4.5.3.1.2.6.3

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

$x < 0$

Step 5.1.4.5.3.1.2.6.4

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

$- x \leq 7$

Step 5.1.4.5.3.1.2.6.5

Write as a piecewise.

${\begin{cases} x \leq 7 & x \geq 0 \\ - x \leq 7 & x < 0 \end{cases}$

Step 5.1.4.5.3.1.2.7

Find the intersection of $x \leq 7$ and $x \geq 0$ .

$0 \leq x \leq 7$

Step 5.1.4.5.3.1.2.8

Solve $- x \leq 7$ when $x < 0$ .

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Step 5.1.4.5.3.1.2.8.1

Divide each term in $- x \leq 7$ by $- 1$ and simplify.

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Step 5.1.4.5.3.1.2.8.1.1

Divide each term in $- x \leq 7$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \geq \frac{7}{- 1}$

Step 5.1.4.5.3.1.2.8.1.2

Simplify the left side.

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Step 5.1.4.5.3.1.2.8.1.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} \geq \frac{7}{- 1}$

Step 5.1.4.5.3.1.2.8.1.2.2

Divide $x$ by $1$ .

$x \geq \frac{7}{- 1}$

Step 5.1.4.5.3.1.2.8.1.3

Simplify the right side.

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Step 5.1.4.5.3.1.2.8.1.3.1

Divide $7$ by $- 1$ .

$x \geq - 7$

Step 5.1.4.5.3.1.2.8.2

Find the intersection of $x \geq - 7$ and $x < 0$ .

$- 7 \leq x < 0$

Step 5.1.4.5.3.1.2.9

Find the union of the solutions.

$- 7 \leq x \leq 7$

Step 5.1.4.5.3.1.3

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

$[- 7, 7]$

Step 5.1.4.5.3.2

Find the intersection of $y \geq 0$ and $[- 7, 7]$ .

$0 \leq y \leq 7$

Step 5.1.4.5.4

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

$y < 0$

Step 5.1.4.5.5

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

$- y > \sqrt{(7 + x) (7 - x)}$

Step 5.1.4.5.6

Find the domain of $- y > \sqrt{(7 + x) (7 - x)}$ and find the intersection with $y < 0$ .

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Step 5.1.4.5.6.1

Find the domain of $- y > \sqrt{(7 + x) (7 - x)}$ .

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Step 5.1.4.5.6.1.1

Set the radicand in $\sqrt{(7 + x) (7 - x)}$ greater than or equal to $0$ to find where the expression is defined.

$(7 + x) (7 - x) \geq 0$

Step 5.1.4.5.6.1.2

Solve for $y$ .

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Step 5.1.4.5.6.1.2.1

Simplify $(7 + x) (7 - x)$ .

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Step 5.1.4.5.6.1.2.1.1

Expand $(7 + x) (7 - x)$ using the FOIL Method.

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Step 5.1.4.5.6.1.2.1.1.1

Apply the distributive property.

$7 (7 - x) + x (7 - x) \geq 0$

Step 5.1.4.5.6.1.2.1.1.2

Apply the distributive property.

$7 \cdot 7 + 7 (- x) + x (7 - x) \geq 0$

Step 5.1.4.5.6.1.2.1.1.3

Apply the distributive property.

$7 \cdot 7 + 7 (- x) + x \cdot 7 + x (- x) \geq 0$

Step 5.1.4.5.6.1.2.1.2

Simplify and combine like terms.

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Step 5.1.4.5.6.1.2.1.2.1

Simplify each term.

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Step 5.1.4.5.6.1.2.1.2.1.1

Multiply $7$ by $7$ .

$49 + 7 (- x) + x \cdot 7 + x (- x) \geq 0$

Step 5.1.4.5.6.1.2.1.2.1.2

Multiply $- 1$ by $7$ .

$49 - 7 x + x \cdot 7 + x (- x) \geq 0$

Step 5.1.4.5.6.1.2.1.2.1.3

Move $7$ to the left of $x$ .

$49 - 7 x + 7 \cdot x + x (- x) \geq 0$

Step 5.1.4.5.6.1.2.1.2.1.4

Rewrite using the commutative property of multiplication.

$49 - 7 x + 7 x - x \cdot x \geq 0$

Step 5.1.4.5.6.1.2.1.2.1.5

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

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Step 5.1.4.5.6.1.2.1.2.1.5.1

Move $x$ .

$49 - 7 x + 7 x - (x \cdot x) \geq 0$

Step 5.1.4.5.6.1.2.1.2.1.5.2

Multiply $x$ by $x$ .

$49 - 7 x + 7 x - x^{2} \geq 0$

Step 5.1.4.5.6.1.2.1.2.2

Add $- 7 x$ and $7 x$ .

$49 + 0 - x^{2} \geq 0$

Step 5.1.4.5.6.1.2.1.2.3

Add $49$ and $0$ .

$49 - x^{2} \geq 0$

Step 5.1.4.5.6.1.2.2

Subtract $49$ from both sides of the inequality.

$- x^{2} \geq - 49$

Step 5.1.4.5.6.1.2.3

Divide each term in $- x^{2} \geq - 49$ by $- 1$ and simplify.

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Step 5.1.4.5.6.1.2.3.1

Divide each term in $- x^{2} \geq - 49$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x^{2}}{- 1} \leq \frac{- 49}{- 1}$

Step 5.1.4.5.6.1.2.3.2

Simplify the left side.

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Step 5.1.4.5.6.1.2.3.2.1

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} \leq \frac{- 49}{- 1}$

Step 5.1.4.5.6.1.2.3.2.2

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

$x^{2} \leq \frac{- 49}{- 1}$

Step 5.1.4.5.6.1.2.3.3

Simplify the right side.

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Step 5.1.4.5.6.1.2.3.3.1

Divide $- 49$ by $- 1$ .

$x^{2} \leq 49$

Step 5.1.4.5.6.1.2.4

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

$\sqrt{x^{2}} \leq \sqrt{49}$

Step 5.1.4.5.6.1.2.5

Simplify the equation.

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Step 5.1.4.5.6.1.2.5.1

Simplify the left side.

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Step 5.1.4.5.6.1.2.5.1.1

Pull terms out from under the radical.

$| x | \leq \sqrt{49}$

Step 5.1.4.5.6.1.2.5.2

Simplify the right side.

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Step 5.1.4.5.6.1.2.5.2.1

Simplify $\sqrt{49}$ .

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Step 5.1.4.5.6.1.2.5.2.1.1

Rewrite $49$ as $7^{2}$ .

$| x | \leq \sqrt{7^{2}}$

Step 5.1.4.5.6.1.2.5.2.1.2

Pull terms out from under the radical.

$| x | \leq | 7 |$

Step 5.1.4.5.6.1.2.5.2.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $7$ is $7$ .

$| x | \leq 7$

Step 5.1.4.5.6.1.2.6

Write $| x | \leq 7$ as a piecewise.

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Step 5.1.4.5.6.1.2.6.1

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

$x \geq 0$

Step 5.1.4.5.6.1.2.6.2

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

$x \leq 7$

Step 5.1.4.5.6.1.2.6.3

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

$x < 0$

Step 5.1.4.5.6.1.2.6.4

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

$- x \leq 7$

Step 5.1.4.5.6.1.2.6.5

Write as a piecewise.

${\begin{cases} x \leq 7 & x \geq 0 \\ - x \leq 7 & x < 0 \end{cases}$

Step 5.1.4.5.6.1.2.7

Find the intersection of $x \leq 7$ and $x \geq 0$ .

$0 \leq x \leq 7$

Step 5.1.4.5.6.1.2.8

Solve $- x \leq 7$ when $x < 0$ .

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Step 5.1.4.5.6.1.2.8.1

Divide each term in $- x \leq 7$ by $- 1$ and simplify.

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Step 5.1.4.5.6.1.2.8.1.1

Divide each term in $- x \leq 7$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \geq \frac{7}{- 1}$

Step 5.1.4.5.6.1.2.8.1.2

Simplify the left side.

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Step 5.1.4.5.6.1.2.8.1.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} \geq \frac{7}{- 1}$

Step 5.1.4.5.6.1.2.8.1.2.2

Divide $x$ by $1$ .

$x \geq \frac{7}{- 1}$

Step 5.1.4.5.6.1.2.8.1.3

Simplify the right side.

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Step 5.1.4.5.6.1.2.8.1.3.1

Divide $7$ by $- 1$ .

$x \geq - 7$

Step 5.1.4.5.6.1.2.8.2

Find the intersection of $x \geq - 7$ and $x < 0$ .

$- 7 \leq x < 0$

Step 5.1.4.5.6.1.2.9

Find the union of the solutions.

$- 7 \leq x \leq 7$

Step 5.1.4.5.6.1.3

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

$[- 7, 7]$

Step 5.1.4.5.6.2

Find the intersection of $y < 0$ and $[- 7, 7]$ .

$- 7 \leq y < 0$

Step 5.1.4.5.7

Write as a piecewise.

${\begin{cases} y > \sqrt{(7 + x) (7 - x)} & 0 \leq y \leq 7 \\ - y > \sqrt{(7 + x) (7 - x)} & - 7 \leq y < 0 \end{cases}$

Step 5.1.4.6

Find the intersection of $y > \sqrt{(7 + x) (7 - x)}$ and $0 \leq y \leq 7$ .

No solution

Step 5.1.4.7

Solve $- y > \sqrt{(7 + x) (7 - x)}$ when $- 7 \leq y < 0$ .

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Step 5.1.4.7.1

Divide each term in $- y > \sqrt{(7 + x) (7 - x)}$ by $- 1$ and simplify.

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Step 5.1.4.7.1.1

Divide each term in $- y > \sqrt{(7 + x) (7 - x)}$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- y}{- 1} < \frac{\sqrt{(7 + x) (7 - x)}}{- 1}$

Step 5.1.4.7.1.2

Simplify the left side.

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Step 5.1.4.7.1.2.1

Dividing two negative values results in a positive value.

$\frac{y}{1} < \frac{\sqrt{(7 + x) (7 - x)}}{- 1}$

Step 5.1.4.7.1.2.2

Divide $y$ by $1$ .

$y < \frac{\sqrt{(7 + x) (7 - x)}}{- 1}$

Step 5.1.4.7.1.3

Simplify the right side.

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Step 5.1.4.7.1.3.1

Move the negative one from the denominator of $\frac{\sqrt{(7 + x) (7 - x)}}{- 1}$ .

$y < - 1 \cdot \sqrt{(7 + x) (7 - x)}$

Step 5.1.4.7.1.3.2

Rewrite $- 1 \cdot \sqrt{(7 + x) (7 - x)}$ as $- \sqrt{(7 + x) (7 - x)}$ .

$y < - \sqrt{(7 + x) (7 - x)}$

Step 5.1.4.7.2

Find the intersection of $y < - \sqrt{(7 + x) (7 - x)}$ and $- 7 \leq y < 0$ .

$- 7 \leq y < N o (Min i m u m)$

Step 5.1.4.8

Find the union of the solutions.

$- 7 \leq y < i N o Min m^{2} u$

Step 5.2

Find the intersection of $- 7 \leq y < i N o Min m^{2} u$ and $0 \leq x \leq 7$ .

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

Step 6

Solve $- x > \sqrt{(7 + y) (7 - y)}$ when $- 7 \leq x < 0$ .

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Step 6.1

Solve $- x > \sqrt{(7 + y) (7 - y)}$ for $y$ .

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Step 6.1.1

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

$\sqrt{(7 + y) (7 - y)} < - x$

Step 6.1.2

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

${\sqrt{(7 + y) (7 - y)}}^{2} < {(- x)}^{2}$

Step 6.1.3

Simplify each side of the inequality.

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Step 6.1.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{(7 + y) (7 - y)}$ as ${((7 + y) (7 - y))}^{\frac{1}{2}}$ .

${({((7 + y) (7 - y))}^{\frac{1}{2}})}^{2} < {(- x)}^{2}$

Step 6.1.3.2

Simplify the left side.

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Step 6.1.3.2.1

Simplify ${({((7 + y) (7 - y))}^{\frac{1}{2}})}^{2}$ .

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Step 6.1.3.2.1.1

Multiply the exponents in ${({((7 + y) (7 - y))}^{\frac{1}{2}})}^{2}$ .

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Step 6.1.3.2.1.1.1

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

${((7 + y) (7 - y))}^{\frac{1}{2} \cdot 2} < {(- x)}^{2}$

Step 6.1.3.2.1.1.2

Cancel the common factor of $2$ .

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Step 6.1.3.2.1.1.2.1

Cancel the common factor.

${((7 + y) (7 - y))}^{\frac{1}{2} \cdot 2} < {(- x)}^{2}$

Step 6.1.3.2.1.1.2.2

Rewrite the expression.

${((7 + y) (7 - y))}^{1} < {(- x)}^{2}$

Step 6.1.3.2.1.2

Expand $(7 + y) (7 - y)$ using the FOIL Method.

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Step 6.1.3.2.1.2.1

Apply the distributive property.

${(7 (7 - y) + y (7 - y))}^{1} < {(- x)}^{2}$

Step 6.1.3.2.1.2.2

Apply the distributive property.

${(7 \cdot 7 + 7 (- y) + y (7 - y))}^{1} < {(- x)}^{2}$

Step 6.1.3.2.1.2.3

Apply the distributive property.

${(7 \cdot 7 + 7 (- y) + y \cdot 7 + y (- y))}^{1} < {(- x)}^{2}$

Step 6.1.3.2.1.3

Simplify and combine like terms.

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Step 6.1.3.2.1.3.1

Simplify each term.

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Step 6.1.3.2.1.3.1.1

Multiply $7$ by $7$ .

${(49 + 7 (- y) + y \cdot 7 + y (- y))}^{1} < {(- x)}^{2}$

Step 6.1.3.2.1.3.1.2

Multiply $- 1$ by $7$ .

${(49 - 7 y + y \cdot 7 + y (- y))}^{1} < {(- x)}^{2}$

Step 6.1.3.2.1.3.1.3

Move $7$ to the left of $y$ .

${(49 - 7 y + 7 \cdot y + y (- y))}^{1} < {(- x)}^{2}$

Step 6.1.3.2.1.3.1.4

Rewrite using the commutative property of multiplication.

${(49 - 7 y + 7 y - y \cdot y)}^{1} < {(- x)}^{2}$

Step 6.1.3.2.1.3.1.5

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

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Step 6.1.3.2.1.3.1.5.1

Move $y$ .

${(49 - 7 y + 7 y - (y \cdot y))}^{1} < {(- x)}^{2}$

Step 6.1.3.2.1.3.1.5.2

Multiply $y$ by $y$ .

${(49 - 7 y + 7 y - y^{2})}^{1} < {(- x)}^{2}$

Step 6.1.3.2.1.3.2

Add $- 7 y$ and $7 y$ .

${(49 + 0 - y^{2})}^{1} < {(- x)}^{2}$

Step 6.1.3.2.1.3.3

Add $49$ and $0$ .

${(49 - y^{2})}^{1} < {(- x)}^{2}$

Step 6.1.3.2.1.4

Simplify.

$49 - y^{2} < {(- x)}^{2}$

Step 6.1.3.3

Simplify the right side.

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Step 6.1.3.3.1

Simplify ${(- x)}^{2}$ .

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Step 6.1.3.3.1.1

Apply the product rule to $- x$ .

$49 - y^{2} < {(- 1)}^{2} x^{2}$

Step 6.1.3.3.1.2

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

$49 - y^{2} < 1 x^{2}$

Step 6.1.3.3.1.3

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

$49 - y^{2} < x^{2}$

Step 6.1.4

Solve for $y$ .

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Step 6.1.4.1

Subtract $49$ from both sides of the inequality.

$- y^{2} < x^{2} - 49$

Step 6.1.4.2

Divide each term in $- y^{2} < x^{2} - 49$ by $- 1$ and simplify.

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Step 6.1.4.2.1

Divide each term in $- y^{2} < x^{2} - 49$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- y^{2}}{- 1} > \frac{x^{2}}{- 1} + \frac{- 49}{- 1}$

Step 6.1.4.2.2

Simplify the left side.

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Step 6.1.4.2.2.1

Dividing two negative values results in a positive value.

$\frac{y^{2}}{1} > \frac{x^{2}}{- 1} + \frac{- 49}{- 1}$

Step 6.1.4.2.2.2

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

$y^{2} > \frac{x^{2}}{- 1} + \frac{- 49}{- 1}$

Step 6.1.4.2.3

Simplify the right side.

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Step 6.1.4.2.3.1

Simplify each term.

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Step 6.1.4.2.3.1.1

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

$y^{2} > - 1 \cdot x^{2} + \frac{- 49}{- 1}$

Step 6.1.4.2.3.1.2

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

$y^{2} > - x^{2} + \frac{- 49}{- 1}$

Step 6.1.4.2.3.1.3

Divide $- 49$ by $- 1$ .

$y^{2} > - x^{2} + 49$

Step 6.1.4.3

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

$\sqrt{y^{2}} > \sqrt{- x^{2} + 49}$

Step 6.1.4.4

Simplify the equation.

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Step 6.1.4.4.1

Simplify the left side.

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Step 6.1.4.4.1.1

Pull terms out from under the radical.

$| y | > \sqrt{- x^{2} + 49}$

Step 6.1.4.4.2

Simplify the right side.

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Step 6.1.4.4.2.1

Simplify $\sqrt{- x^{2} + 49}$ .

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Step 6.1.4.4.2.1.1

Simplify the expression.

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Step 6.1.4.4.2.1.1.1

Rewrite $49$ as $7^{2}$ .

$| y | > \sqrt{- x^{2} + 7^{2}}$

Step 6.1.4.4.2.1.1.2

Reorder $- x^{2}$ and $7^{2}$ .

$| y | > \sqrt{7^{2} - x^{2}}$

Step 6.1.4.4.2.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 7$ and $b = x$ .

$| y | > \sqrt{(7 + x) (7 - x)}$

Step 6.1.4.5

Write $| y | > \sqrt{(7 + x) (7 - x)}$ as a piecewise.

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Step 6.1.4.5.1

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

$y \geq 0$

Step 6.1.4.5.2

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

$y > \sqrt{(7 + x) (7 - x)}$

Step 6.1.4.5.3

Find the domain of $y > \sqrt{(7 + x) (7 - x)}$ and find the intersection with $y \geq 0$ .

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Step 6.1.4.5.3.1

Find the domain of $y > \sqrt{(7 + x) (7 - x)}$ .

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Step 6.1.4.5.3.1.1

Set the radicand in $\sqrt{(7 + x) (7 - x)}$ greater than or equal to $0$ to find where the expression is defined.

$(7 + x) (7 - x) \geq 0$

Step 6.1.4.5.3.1.2

Solve for $y$ .

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Step 6.1.4.5.3.1.2.1

Simplify $(7 + x) (7 - x)$ .

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Step 6.1.4.5.3.1.2.1.1

Expand $(7 + x) (7 - x)$ using the FOIL Method.

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Step 6.1.4.5.3.1.2.1.1.1

Apply the distributive property.

$7 (7 - x) + x (7 - x) \geq 0$

Step 6.1.4.5.3.1.2.1.1.2

Apply the distributive property.

$7 \cdot 7 + 7 (- x) + x (7 - x) \geq 0$

Step 6.1.4.5.3.1.2.1.1.3

Apply the distributive property.

$7 \cdot 7 + 7 (- x) + x \cdot 7 + x (- x) \geq 0$

Step 6.1.4.5.3.1.2.1.2

Simplify and combine like terms.

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Step 6.1.4.5.3.1.2.1.2.1

Simplify each term.

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Step 6.1.4.5.3.1.2.1.2.1.1

Multiply $7$ by $7$ .

$49 + 7 (- x) + x \cdot 7 + x (- x) \geq 0$

Step 6.1.4.5.3.1.2.1.2.1.2

Multiply $- 1$ by $7$ .

$49 - 7 x + x \cdot 7 + x (- x) \geq 0$

Step 6.1.4.5.3.1.2.1.2.1.3

Move $7$ to the left of $x$ .

$49 - 7 x + 7 \cdot x + x (- x) \geq 0$

Step 6.1.4.5.3.1.2.1.2.1.4

Rewrite using the commutative property of multiplication.

$49 - 7 x + 7 x - x \cdot x \geq 0$

Step 6.1.4.5.3.1.2.1.2.1.5

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

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Step 6.1.4.5.3.1.2.1.2.1.5.1

Move $x$ .

$49 - 7 x + 7 x - (x \cdot x) \geq 0$

Step 6.1.4.5.3.1.2.1.2.1.5.2

Multiply $x$ by $x$ .

$49 - 7 x + 7 x - x^{2} \geq 0$

Step 6.1.4.5.3.1.2.1.2.2

Add $- 7 x$ and $7 x$ .

$49 + 0 - x^{2} \geq 0$

Step 6.1.4.5.3.1.2.1.2.3

Add $49$ and $0$ .

$49 - x^{2} \geq 0$

Step 6.1.4.5.3.1.2.2

Subtract $49$ from both sides of the inequality.

$- x^{2} \geq - 49$

Step 6.1.4.5.3.1.2.3

Divide each term in $- x^{2} \geq - 49$ by $- 1$ and simplify.

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Step 6.1.4.5.3.1.2.3.1

Divide each term in $- x^{2} \geq - 49$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x^{2}}{- 1} \leq \frac{- 49}{- 1}$

Step 6.1.4.5.3.1.2.3.2

Simplify the left side.

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Step 6.1.4.5.3.1.2.3.2.1

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} \leq \frac{- 49}{- 1}$

Step 6.1.4.5.3.1.2.3.2.2

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

$x^{2} \leq \frac{- 49}{- 1}$

Step 6.1.4.5.3.1.2.3.3

Simplify the right side.

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Step 6.1.4.5.3.1.2.3.3.1

Divide $- 49$ by $- 1$ .

$x^{2} \leq 49$

Step 6.1.4.5.3.1.2.4

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

$\sqrt{x^{2}} \leq \sqrt{49}$

Step 6.1.4.5.3.1.2.5

Simplify the equation.

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Step 6.1.4.5.3.1.2.5.1

Simplify the left side.

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Step 6.1.4.5.3.1.2.5.1.1

Pull terms out from under the radical.

$| x | \leq \sqrt{49}$

Step 6.1.4.5.3.1.2.5.2

Simplify the right side.

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Step 6.1.4.5.3.1.2.5.2.1

Simplify $\sqrt{49}$ .

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Step 6.1.4.5.3.1.2.5.2.1.1

Rewrite $49$ as $7^{2}$ .

$| x | \leq \sqrt{7^{2}}$

Step 6.1.4.5.3.1.2.5.2.1.2

Pull terms out from under the radical.

$| x | \leq | 7 |$

Step 6.1.4.5.3.1.2.5.2.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $7$ is $7$ .

$| x | \leq 7$

Step 6.1.4.5.3.1.2.6

Write $| x | \leq 7$ as a piecewise.

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Step 6.1.4.5.3.1.2.6.1

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

$x \geq 0$

Step 6.1.4.5.3.1.2.6.2

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

$x \leq 7$

Step 6.1.4.5.3.1.2.6.3

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

$x < 0$

Step 6.1.4.5.3.1.2.6.4

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

$- x \leq 7$

Step 6.1.4.5.3.1.2.6.5

Write as a piecewise.

${\begin{cases} x \leq 7 & x \geq 0 \\ - x \leq 7 & x < 0 \end{cases}$

Step 6.1.4.5.3.1.2.7

Find the intersection of $x \leq 7$ and $x \geq 0$ .

$0 \leq x \leq 7$

Step 6.1.4.5.3.1.2.8

Solve $- x \leq 7$ when $x < 0$ .

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Step 6.1.4.5.3.1.2.8.1

Divide each term in $- x \leq 7$ by $- 1$ and simplify.

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Step 6.1.4.5.3.1.2.8.1.1

Divide each term in $- x \leq 7$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \geq \frac{7}{- 1}$

Step 6.1.4.5.3.1.2.8.1.2

Simplify the left side.

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Step 6.1.4.5.3.1.2.8.1.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} \geq \frac{7}{- 1}$

Step 6.1.4.5.3.1.2.8.1.2.2

Divide $x$ by $1$ .

$x \geq \frac{7}{- 1}$

Step 6.1.4.5.3.1.2.8.1.3

Simplify the right side.

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Step 6.1.4.5.3.1.2.8.1.3.1

Divide $7$ by $- 1$ .

$x \geq - 7$

Step 6.1.4.5.3.1.2.8.2

Find the intersection of $x \geq - 7$ and $x < 0$ .

$- 7 \leq x < 0$

Step 6.1.4.5.3.1.2.9

Find the union of the solutions.

$- 7 \leq x \leq 7$

Step 6.1.4.5.3.1.3

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

$[- 7, 7]$

Step 6.1.4.5.3.2

Find the intersection of $y \geq 0$ and $[- 7, 7]$ .

$0 \leq y \leq 7$

Step 6.1.4.5.4

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

$y < 0$

Step 6.1.4.5.5

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

$- y > \sqrt{(7 + x) (7 - x)}$

Step 6.1.4.5.6

Find the domain of $- y > \sqrt{(7 + x) (7 - x)}$ and find the intersection with $y < 0$ .

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Step 6.1.4.5.6.1

Find the domain of $- y > \sqrt{(7 + x) (7 - x)}$ .

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Step 6.1.4.5.6.1.1

Set the radicand in $\sqrt{(7 + x) (7 - x)}$ greater than or equal to $0$ to find where the expression is defined.

$(7 + x) (7 - x) \geq 0$

Step 6.1.4.5.6.1.2

Solve for $y$ .

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Step 6.1.4.5.6.1.2.1

Simplify $(7 + x) (7 - x)$ .

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Step 6.1.4.5.6.1.2.1.1

Expand $(7 + x) (7 - x)$ using the FOIL Method.

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Step 6.1.4.5.6.1.2.1.1.1

Apply the distributive property.

$7 (7 - x) + x (7 - x) \geq 0$

Step 6.1.4.5.6.1.2.1.1.2

Apply the distributive property.

$7 \cdot 7 + 7 (- x) + x (7 - x) \geq 0$

Step 6.1.4.5.6.1.2.1.1.3

Apply the distributive property.

$7 \cdot 7 + 7 (- x) + x \cdot 7 + x (- x) \geq 0$

Step 6.1.4.5.6.1.2.1.2

Simplify and combine like terms.

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Step 6.1.4.5.6.1.2.1.2.1

Simplify each term.

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Step 6.1.4.5.6.1.2.1.2.1.1

Multiply $7$ by $7$ .

$49 + 7 (- x) + x \cdot 7 + x (- x) \geq 0$

Step 6.1.4.5.6.1.2.1.2.1.2

Multiply $- 1$ by $7$ .

$49 - 7 x + x \cdot 7 + x (- x) \geq 0$

Step 6.1.4.5.6.1.2.1.2.1.3

Move $7$ to the left of $x$ .

$49 - 7 x + 7 \cdot x + x (- x) \geq 0$

Step 6.1.4.5.6.1.2.1.2.1.4

Rewrite using the commutative property of multiplication.

$49 - 7 x + 7 x - x \cdot x \geq 0$

Step 6.1.4.5.6.1.2.1.2.1.5

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

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Step 6.1.4.5.6.1.2.1.2.1.5.1

Move $x$ .

$49 - 7 x + 7 x - (x \cdot x) \geq 0$

Step 6.1.4.5.6.1.2.1.2.1.5.2

Multiply $x$ by $x$ .

$49 - 7 x + 7 x - x^{2} \geq 0$

Step 6.1.4.5.6.1.2.1.2.2

Add $- 7 x$ and $7 x$ .

$49 + 0 - x^{2} \geq 0$

Step 6.1.4.5.6.1.2.1.2.3

Add $49$ and $0$ .

$49 - x^{2} \geq 0$

Step 6.1.4.5.6.1.2.2

Subtract $49$ from both sides of the inequality.

$- x^{2} \geq - 49$

Step 6.1.4.5.6.1.2.3

Divide each term in $- x^{2} \geq - 49$ by $- 1$ and simplify.

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Step 6.1.4.5.6.1.2.3.1

Divide each term in $- x^{2} \geq - 49$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x^{2}}{- 1} \leq \frac{- 49}{- 1}$

Step 6.1.4.5.6.1.2.3.2

Simplify the left side.

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Step 6.1.4.5.6.1.2.3.2.1

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} \leq \frac{- 49}{- 1}$

Step 6.1.4.5.6.1.2.3.2.2

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

$x^{2} \leq \frac{- 49}{- 1}$

Step 6.1.4.5.6.1.2.3.3

Simplify the right side.

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Step 6.1.4.5.6.1.2.3.3.1

Divide $- 49$ by $- 1$ .

$x^{2} \leq 49$

Step 6.1.4.5.6.1.2.4

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

$\sqrt{x^{2}} \leq \sqrt{49}$

Step 6.1.4.5.6.1.2.5

Simplify the equation.

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Step 6.1.4.5.6.1.2.5.1

Simplify the left side.

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Step 6.1.4.5.6.1.2.5.1.1

Pull terms out from under the radical.

$| x | \leq \sqrt{49}$

Step 6.1.4.5.6.1.2.5.2

Simplify the right side.

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Step 6.1.4.5.6.1.2.5.2.1

Simplify $\sqrt{49}$ .

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Step 6.1.4.5.6.1.2.5.2.1.1

Rewrite $49$ as $7^{2}$ .

$| x | \leq \sqrt{7^{2}}$

Step 6.1.4.5.6.1.2.5.2.1.2

Pull terms out from under the radical.

$| x | \leq | 7 |$

Step 6.1.4.5.6.1.2.5.2.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $7$ is $7$ .

$| x | \leq 7$

Step 6.1.4.5.6.1.2.6

Write $| x | \leq 7$ as a piecewise.

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Step 6.1.4.5.6.1.2.6.1

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

$x \geq 0$

Step 6.1.4.5.6.1.2.6.2

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

$x \leq 7$

Step 6.1.4.5.6.1.2.6.3

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

$x < 0$

Step 6.1.4.5.6.1.2.6.4

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

$- x \leq 7$

Step 6.1.4.5.6.1.2.6.5

Write as a piecewise.

${\begin{cases} x \leq 7 & x \geq 0 \\ - x \leq 7 & x < 0 \end{cases}$

Step 6.1.4.5.6.1.2.7

Find the intersection of $x \leq 7$ and $x \geq 0$ .

$0 \leq x \leq 7$

Step 6.1.4.5.6.1.2.8

Solve $- x \leq 7$ when $x < 0$ .

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Step 6.1.4.5.6.1.2.8.1

Divide each term in $- x \leq 7$ by $- 1$ and simplify.

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Step 6.1.4.5.6.1.2.8.1.1

Divide each term in $- x \leq 7$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \geq \frac{7}{- 1}$

Step 6.1.4.5.6.1.2.8.1.2

Simplify the left side.

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Step 6.1.4.5.6.1.2.8.1.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} \geq \frac{7}{- 1}$

Step 6.1.4.5.6.1.2.8.1.2.2

Divide $x$ by $1$ .

$x \geq \frac{7}{- 1}$

Step 6.1.4.5.6.1.2.8.1.3

Simplify the right side.

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Step 6.1.4.5.6.1.2.8.1.3.1

Divide $7$ by $- 1$ .

$x \geq - 7$

Step 6.1.4.5.6.1.2.8.2

Find the intersection of $x \geq - 7$ and $x < 0$ .

$- 7 \leq x < 0$

Step 6.1.4.5.6.1.2.9

Find the union of the solutions.

$- 7 \leq x \leq 7$

Step 6.1.4.5.6.1.3

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

$[- 7, 7]$

Step 6.1.4.5.6.2

Find the intersection of $y < 0$ and $[- 7, 7]$ .

$- 7 \leq y < 0$

Step 6.1.4.5.7

Write as a piecewise.

${\begin{cases} y > \sqrt{(7 + x) (7 - x)} & 0 \leq y \leq 7 \\ - y > \sqrt{(7 + x) (7 - x)} & - 7 \leq y < 0 \end{cases}$

Step 6.1.4.6

Find the intersection of $y > \sqrt{(7 + x) (7 - x)}$ and $0 \leq y \leq 7$ .

No solution

Step 6.1.4.7

Solve $- y > \sqrt{(7 + x) (7 - x)}$ when $- 7 \leq y < 0$ .

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Step 6.1.4.7.1

Divide each term in $- y > \sqrt{(7 + x) (7 - x)}$ by $- 1$ and simplify.

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Step 6.1.4.7.1.1

Divide each term in $- y > \sqrt{(7 + x) (7 - x)}$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- y}{- 1} < \frac{\sqrt{(7 + x) (7 - x)}}{- 1}$

Step 6.1.4.7.1.2

Simplify the left side.

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Step 6.1.4.7.1.2.1

Dividing two negative values results in a positive value.

$\frac{y}{1} < \frac{\sqrt{(7 + x) (7 - x)}}{- 1}$

Step 6.1.4.7.1.2.2

Divide $y$ by $1$ .

$y < \frac{\sqrt{(7 + x) (7 - x)}}{- 1}$

Step 6.1.4.7.1.3

Simplify the right side.

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Step 6.1.4.7.1.3.1

Move the negative one from the denominator of $\frac{\sqrt{(7 + x) (7 - x)}}{- 1}$ .

$y < - 1 \cdot \sqrt{(7 + x) (7 - x)}$

Step 6.1.4.7.1.3.2

Rewrite $- 1 \cdot \sqrt{(7 + x) (7 - x)}$ as $- \sqrt{(7 + x) (7 - x)}$ .

$y < - \sqrt{(7 + x) (7 - x)}$

Step 6.1.4.7.2

Find the intersection of $y < - \sqrt{(7 + x) (7 - x)}$ and $- 7 \leq y < 0$ .

$- 7 \leq y < N o (Min i m u m)$

Step 6.1.4.8

Find the union of the solutions.

$- 7 \leq y < i N o Min m^{2} u$

Step 6.2

Find the intersection of $- 7 \leq y < i N o Min m^{2} u$ and $- 7 \leq x < 0$ .

$- 7 \leq x < N o (Min i m u m)$

Step 7

Find the union of the solutions.

$- 7 \leq x < N o (Max i m u m)$