Calculus Examples

$x^{2} + y^{2} + z^{2} - 64 = 0$

Step 1

Move all terms not containing $y$ to the right side of the equation.

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Step 1.1

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

$y^{2} + z^{2} - 64 = - x^{2}$

Step 1.2

Subtract $z^{2}$ from both sides of the equation.

$y^{2} - 64 = - x^{2} - z^{2}$

Step 1.3

Add $64$ to both sides of the equation.

$y^{2} = - x^{2} - z^{2} + 64$

Step 2

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

$y = \pm \sqrt{- x^{2} - z^{2} + 64}$

Step 3

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.1

First, use the positive value of the $\pm$ to find the first solution.

$y = \sqrt{- x^{2} - z^{2} + 64}$

Step 3.2

Next, use the negative value of the $\pm$ to find the second solution.

$y = - \sqrt{- x^{2} - z^{2} + 64}$

Step 3.3

The complete solution is the result of both the positive and negative portions of the solution.

$y = \sqrt{- x^{2} - z^{2} + 64}$

$y = - \sqrt{- x^{2} - z^{2} + 64}$

$y = \sqrt{- x^{2} - z^{2} + 64}$

$y = - \sqrt{- x^{2} - z^{2} + 64}$

Step 4

Set the radicand in $\sqrt{- x^{2} - z^{2} + 64}$ greater than or equal to $0$ to find where the expression is defined.

$- x^{2} - z^{2} + 64 \geq 0$

Step 5

Solve for $x$ .

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Step 5.1

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

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Step 5.1.1

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

$- x^{2} + 64 \geq z^{2}$

Step 5.1.2

Subtract $64$ from both sides of the inequality.

$- x^{2} \geq z^{2} - 64$

Step 5.2

Divide each term in $- x^{2} \geq z^{2} - 64$ by $- 1$ and simplify.

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Step 5.2.1

Divide each term in $- x^{2} \geq z^{2} - 64$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x^{2}}{- 1} \leq \frac{z^{2}}{- 1} + \frac{- 64}{- 1}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} \leq \frac{z^{2}}{- 1} + \frac{- 64}{- 1}$

Step 5.2.2.2

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

$x^{2} \leq \frac{z^{2}}{- 1} + \frac{- 64}{- 1}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Simplify each term.

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Step 5.2.3.1.1

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

$x^{2} \leq - 1 \cdot z^{2} + \frac{- 64}{- 1}$

Step 5.2.3.1.2

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

$x^{2} \leq - z^{2} + \frac{- 64}{- 1}$

Step 5.2.3.1.3

Divide $- 64$ by $- 1$ .

$x^{2} \leq - z^{2} + 64$

Step 5.3

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

$\sqrt{x^{2}} \leq \sqrt{- z^{2} + 64}$

Step 5.4

Simplify the equation.

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Step 5.4.1

Simplify the left side.

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Step 5.4.1.1

Pull terms out from under the radical.

$| x | \leq \sqrt{- z^{2} + 64}$

Step 5.4.2

Simplify the right side.

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Step 5.4.2.1

Simplify $\sqrt{- z^{2} + 64}$ .

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Step 5.4.2.1.1

Simplify the expression.

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Step 5.4.2.1.1.1

Rewrite $64$ as $8^{2}$ .

$| x | \leq \sqrt{- z^{2} + 8^{2}}$

Step 5.4.2.1.1.2

Reorder $- z^{2}$ and $8^{2}$ .

$| x | \leq \sqrt{8^{2} - z^{2}}$

Step 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 $a = 8$ and $b = z$ .

$| x | \leq \sqrt{(8 + z) (8 - z)}$

Step 5.5

Write $| x | \leq \sqrt{(8 + z) (8 - z)}$ as a piecewise.

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Step 5.5.1

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

$x \geq 0$

Step 5.5.2

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

$x \leq \sqrt{(8 + z) (8 - z)}$

Step 5.5.3

Find the domain of $x \leq \sqrt{(8 + z) (8 - z)}$ and find the intersection with $x \geq 0$ .

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Step 5.5.3.1

Find the domain of $x \leq \sqrt{(8 + z) (8 - z)}$ .

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Step 5.5.3.1.1

Set the radicand in $\sqrt{(8 + z) (8 - z)}$ greater than or equal to $0$ to find where the expression is defined.

$(8 + z) (8 - z) \geq 0$

Step 5.5.3.1.2

Solve for $x$ .

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Step 5.5.3.1.2.1

Simplify $(8 + z) (8 - z)$ .

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Step 5.5.3.1.2.1.1

Expand $(8 + z) (8 - z)$ using the FOIL Method.

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Step 5.5.3.1.2.1.1.1

Apply the distributive property.

$8 (8 - z) + z (8 - z) \geq 0$

Step 5.5.3.1.2.1.1.2

Apply the distributive property.

$8 \cdot 8 + 8 (- z) + z (8 - z) \geq 0$

Step 5.5.3.1.2.1.1.3

Apply the distributive property.

$8 \cdot 8 + 8 (- z) + z \cdot 8 + z (- z) \geq 0$

Step 5.5.3.1.2.1.2

Simplify and combine like terms.

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Step 5.5.3.1.2.1.2.1

Simplify each term.

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Step 5.5.3.1.2.1.2.1.1

Multiply $8$ by $8$ .

$64 + 8 (- z) + z \cdot 8 + z (- z) \geq 0$

Step 5.5.3.1.2.1.2.1.2

Multiply $- 1$ by $8$ .

$64 - 8 z + z \cdot 8 + z (- z) \geq 0$

Step 5.5.3.1.2.1.2.1.3

Move $8$ to the left of $z$ .

$64 - 8 z + 8 \cdot z + z (- z) \geq 0$

Step 5.5.3.1.2.1.2.1.4

Rewrite using the commutative property of multiplication.

$64 - 8 z + 8 z - z \cdot z \geq 0$

Step 5.5.3.1.2.1.2.1.5

Multiply $z$ by $z$ by adding the exponents.

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Step 5.5.3.1.2.1.2.1.5.1

Move $z$ .

$64 - 8 z + 8 z - (z \cdot z) \geq 0$

Step 5.5.3.1.2.1.2.1.5.2

Multiply $z$ by $z$ .

$64 - 8 z + 8 z - z^{2} \geq 0$

Step 5.5.3.1.2.1.2.2

Add $- 8 z$ and $8 z$ .

$64 + 0 - z^{2} \geq 0$

Step 5.5.3.1.2.1.2.3

Add $64$ and $0$ .

$64 - z^{2} \geq 0$

Step 5.5.3.1.2.2

Subtract $64$ from both sides of the inequality.

$- z^{2} \geq - 64$

Step 5.5.3.1.2.3

Divide each term in $- z^{2} \geq - 64$ by $- 1$ and simplify.

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Step 5.5.3.1.2.3.1

Divide each term in $- z^{2} \geq - 64$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- z^{2}}{- 1} \leq \frac{- 64}{- 1}$

Step 5.5.3.1.2.3.2

Simplify the left side.

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Step 5.5.3.1.2.3.2.1

Dividing two negative values results in a positive value.

$\frac{z^{2}}{1} \leq \frac{- 64}{- 1}$

Step 5.5.3.1.2.3.2.2

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

$z^{2} \leq \frac{- 64}{- 1}$

Step 5.5.3.1.2.3.3

Simplify the right side.

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Step 5.5.3.1.2.3.3.1

Divide $- 64$ by $- 1$ .

$z^{2} \leq 64$

Step 5.5.3.1.2.4

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

$\sqrt{z^{2}} \leq \sqrt{64}$

Step 5.5.3.1.2.5

Simplify the equation.

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Step 5.5.3.1.2.5.1

Simplify the left side.

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Step 5.5.3.1.2.5.1.1

Pull terms out from under the radical.

$| z | \leq \sqrt{64}$

Step 5.5.3.1.2.5.2

Simplify the right side.

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Step 5.5.3.1.2.5.2.1

Simplify $\sqrt{64}$ .

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Step 5.5.3.1.2.5.2.1.1

Rewrite $64$ as $8^{2}$ .

$| z | \leq \sqrt{8^{2}}$

Step 5.5.3.1.2.5.2.1.2

Pull terms out from under the radical.

$| z | \leq | 8 |$

Step 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 $8$ is $8$ .

$| z | \leq 8$

Step 5.5.3.1.2.6

Write $| z | \leq 8$ as a piecewise.

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Step 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.

$z \geq 0$

Step 5.5.3.1.2.6.2

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

$z \leq 8$

Step 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.

$z < 0$

Step 5.5.3.1.2.6.4

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

$- z \leq 8$

Step 5.5.3.1.2.6.5

Write as a piecewise.

${\begin{cases} z \leq 8 & z \geq 0 \\ - z \leq 8 & z < 0 \end{cases}$

Step 5.5.3.1.2.7

Find the intersection of $z \leq 8$ and $z \geq 0$ .

$0 \leq z \leq 8$

Step 5.5.3.1.2.8

Solve $- z \leq 8$ when $z < 0$ .

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Step 5.5.3.1.2.8.1

Divide each term in $- z \leq 8$ by $- 1$ and simplify.

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Step 5.5.3.1.2.8.1.1

Divide each term in $- z \leq 8$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- z}{- 1} \geq \frac{8}{- 1}$

Step 5.5.3.1.2.8.1.2

Simplify the left side.

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Step 5.5.3.1.2.8.1.2.1

Dividing two negative values results in a positive value.

$\frac{z}{1} \geq \frac{8}{- 1}$

Step 5.5.3.1.2.8.1.2.2

Divide $z$ by $1$ .

$z \geq \frac{8}{- 1}$

Step 5.5.3.1.2.8.1.3

Simplify the right side.

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Step 5.5.3.1.2.8.1.3.1

Divide $8$ by $- 1$ .

$z \geq - 8$

Step 5.5.3.1.2.8.2

Find the intersection of $z \geq - 8$ and $z < 0$ .

$- 8 \leq z < 0$

Step 5.5.3.1.2.9

Find the union of the solutions.

$- 8 \leq z \leq 8$

Step 5.5.3.1.3

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

$[- 8, 8]$

Step 5.5.3.2

Find the intersection of $x \geq 0$ and $[- 8, 8]$ .

$0 \leq x \leq 8$

Step 5.5.4

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

$x < 0$

Step 5.5.5

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

$- x \leq \sqrt{(8 + z) (8 - z)}$

Step 5.5.6

Find the domain of $- x \leq \sqrt{(8 + z) (8 - z)}$ and find the intersection with $x < 0$ .

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Step 5.5.6.1

Find the domain of $- x \leq \sqrt{(8 + z) (8 - z)}$ .

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Step 5.5.6.1.1

Set the radicand in $\sqrt{(8 + z) (8 - z)}$ greater than or equal to $0$ to find where the expression is defined.

$(8 + z) (8 - z) \geq 0$

Step 5.5.6.1.2

Solve for $x$ .

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Step 5.5.6.1.2.1

Simplify $(8 + z) (8 - z)$ .

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Step 5.5.6.1.2.1.1

Expand $(8 + z) (8 - z)$ using the FOIL Method.

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Step 5.5.6.1.2.1.1.1

Apply the distributive property.

$8 (8 - z) + z (8 - z) \geq 0$

Step 5.5.6.1.2.1.1.2

Apply the distributive property.

$8 \cdot 8 + 8 (- z) + z (8 - z) \geq 0$

Step 5.5.6.1.2.1.1.3

Apply the distributive property.

$8 \cdot 8 + 8 (- z) + z \cdot 8 + z (- z) \geq 0$

Step 5.5.6.1.2.1.2

Simplify and combine like terms.

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Step 5.5.6.1.2.1.2.1

Simplify each term.

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Step 5.5.6.1.2.1.2.1.1

Multiply $8$ by $8$ .

$64 + 8 (- z) + z \cdot 8 + z (- z) \geq 0$

Step 5.5.6.1.2.1.2.1.2

Multiply $- 1$ by $8$ .

$64 - 8 z + z \cdot 8 + z (- z) \geq 0$

Step 5.5.6.1.2.1.2.1.3

Move $8$ to the left of $z$ .

$64 - 8 z + 8 \cdot z + z (- z) \geq 0$

Step 5.5.6.1.2.1.2.1.4

Rewrite using the commutative property of multiplication.

$64 - 8 z + 8 z - z \cdot z \geq 0$

Step 5.5.6.1.2.1.2.1.5

Multiply $z$ by $z$ by adding the exponents.

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Step 5.5.6.1.2.1.2.1.5.1

Move $z$ .

$64 - 8 z + 8 z - (z \cdot z) \geq 0$

Step 5.5.6.1.2.1.2.1.5.2

Multiply $z$ by $z$ .

$64 - 8 z + 8 z - z^{2} \geq 0$

Step 5.5.6.1.2.1.2.2

Add $- 8 z$ and $8 z$ .

$64 + 0 - z^{2} \geq 0$

Step 5.5.6.1.2.1.2.3

Add $64$ and $0$ .

$64 - z^{2} \geq 0$

Step 5.5.6.1.2.2

Subtract $64$ from both sides of the inequality.

$- z^{2} \geq - 64$

Step 5.5.6.1.2.3

Divide each term in $- z^{2} \geq - 64$ by $- 1$ and simplify.

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Step 5.5.6.1.2.3.1

Divide each term in $- z^{2} \geq - 64$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- z^{2}}{- 1} \leq \frac{- 64}{- 1}$

Step 5.5.6.1.2.3.2

Simplify the left side.

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Step 5.5.6.1.2.3.2.1

Dividing two negative values results in a positive value.

$\frac{z^{2}}{1} \leq \frac{- 64}{- 1}$

Step 5.5.6.1.2.3.2.2

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

$z^{2} \leq \frac{- 64}{- 1}$

Step 5.5.6.1.2.3.3

Simplify the right side.

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Step 5.5.6.1.2.3.3.1

Divide $- 64$ by $- 1$ .

$z^{2} \leq 64$

Step 5.5.6.1.2.4

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

$\sqrt{z^{2}} \leq \sqrt{64}$

Step 5.5.6.1.2.5

Simplify the equation.

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Step 5.5.6.1.2.5.1

Simplify the left side.

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Step 5.5.6.1.2.5.1.1

Pull terms out from under the radical.

$| z | \leq \sqrt{64}$

Step 5.5.6.1.2.5.2

Simplify the right side.

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Step 5.5.6.1.2.5.2.1

Simplify $\sqrt{64}$ .

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Step 5.5.6.1.2.5.2.1.1

Rewrite $64$ as $8^{2}$ .

$| z | \leq \sqrt{8^{2}}$

Step 5.5.6.1.2.5.2.1.2

Pull terms out from under the radical.

$| z | \leq | 8 |$

Step 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 $8$ is $8$ .

$| z | \leq 8$

Step 5.5.6.1.2.6

Write $| z | \leq 8$ as a piecewise.

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Step 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.

$z \geq 0$

Step 5.5.6.1.2.6.2

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

$z \leq 8$

Step 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.

$z < 0$

Step 5.5.6.1.2.6.4

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

$- z \leq 8$

Step 5.5.6.1.2.6.5

Write as a piecewise.

${\begin{cases} z \leq 8 & z \geq 0 \\ - z \leq 8 & z < 0 \end{cases}$

Step 5.5.6.1.2.7

Find the intersection of $z \leq 8$ and $z \geq 0$ .

$0 \leq z \leq 8$

Step 5.5.6.1.2.8

Solve $- z \leq 8$ when $z < 0$ .

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Step 5.5.6.1.2.8.1

Divide each term in $- z \leq 8$ by $- 1$ and simplify.

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Step 5.5.6.1.2.8.1.1

Divide each term in $- z \leq 8$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- z}{- 1} \geq \frac{8}{- 1}$

Step 5.5.6.1.2.8.1.2

Simplify the left side.

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Step 5.5.6.1.2.8.1.2.1

Dividing two negative values results in a positive value.

$\frac{z}{1} \geq \frac{8}{- 1}$

Step 5.5.6.1.2.8.1.2.2

Divide $z$ by $1$ .

$z \geq \frac{8}{- 1}$

Step 5.5.6.1.2.8.1.3

Simplify the right side.

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Step 5.5.6.1.2.8.1.3.1

Divide $8$ by $- 1$ .

$z \geq - 8$

Step 5.5.6.1.2.8.2

Find the intersection of $z \geq - 8$ and $z < 0$ .

$- 8 \leq z < 0$

Step 5.5.6.1.2.9

Find the union of the solutions.

$- 8 \leq z \leq 8$

Step 5.5.6.1.3

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

$[- 8, 8]$

Step 5.5.6.2

Find the intersection of $x < 0$ and $[- 8, 8]$ .

$- 8 \leq x < 0$

Step 5.5.7

Write as a piecewise.

${\begin{cases} x \leq \sqrt{(8 + z) (8 - z)} & 0 \leq x \leq 8 \\ - x \leq \sqrt{(8 + z) (8 - z)} & - 8 \leq x < 0 \end{cases}$

Step 5.6

Find the intersection of $x \leq \sqrt{(8 + z) (8 - z)}$ and $0 \leq x \leq 8$ .

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

Step 5.7

Solve $- x \leq \sqrt{(8 + z) (8 - z)}$ when $- 8 \leq x < 0$ .

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Step 5.7.1

Divide each term in $- x \leq \sqrt{(8 + z) (8 - z)}$ by $- 1$ and simplify.

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Step 5.7.1.1

Divide each term in $- x \leq \sqrt{(8 + z) (8 - z)}$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \geq \frac{\sqrt{(8 + z) (8 - z)}}{- 1}$

Step 5.7.1.2

Simplify the left side.

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Step 5.7.1.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} \geq \frac{\sqrt{(8 + z) (8 - z)}}{- 1}$

Step 5.7.1.2.2

Divide $x$ by $1$ .

$x \geq \frac{\sqrt{(8 + z) (8 - z)}}{- 1}$

Step 5.7.1.3

Simplify the right side.

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Step 5.7.1.3.1

Move the negative one from the denominator of $\frac{\sqrt{(8 + z) (8 - z)}}{- 1}$ .

$x \geq - 1 \cdot \sqrt{(8 + z) (8 - z)}$

Step 5.7.1.3.2

Rewrite $- 1 \cdot \sqrt{(8 + z) (8 - z)}$ as $- \sqrt{(8 + z) (8 - z)}$ .

$x \geq - \sqrt{(8 + z) (8 - z)}$

Step 5.7.2

Find the intersection of $x \geq - \sqrt{(8 + z) (8 - z)}$ and $- 8 \leq x < 0$ .

No solution

Step 5.8

Find the union of the solutions.

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

Step 6

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