Finite Math Examples

$9 x^{2} - 4 y^{2} + 36 z^{2} = 36$

Step 1

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

Tap for more steps...

Step 1.1

Subtract $9 x^{2}$ from both sides of the equation.

$- 4 y^{2} + 36 z^{2} = 36 - 9 x^{2}$

Step 1.2

Subtract $36 z^{2}$ from both sides of the equation.

$- 4 y^{2} = 36 - 9 x^{2} - 36 z^{2}$

Step 2

Divide each term in $- 4 y^{2} = 36 - 9 x^{2} - 36 z^{2}$ by $- 4$ and simplify.

Tap for more steps...

Step 2.1

Divide each term in $- 4 y^{2} = 36 - 9 x^{2} - 36 z^{2}$ by $- 4$ .

$\frac{- 4 y^{2}}{- 4} = \frac{36}{- 4} + \frac{- 9 x^{2}}{- 4} + \frac{- 36 z^{2}}{- 4}$

Step 2.2

Simplify the left side.

Tap for more steps...

Step 2.2.1

Cancel the common factor of $- 4$ .

Tap for more steps...

Step 2.2.1.1

Cancel the common factor.

$\frac{- 4 y^{2}}{- 4} = \frac{36}{- 4} + \frac{- 9 x^{2}}{- 4} + \frac{- 36 z^{2}}{- 4}$

Step 2.2.1.2

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

$y^{2} = \frac{36}{- 4} + \frac{- 9 x^{2}}{- 4} + \frac{- 36 z^{2}}{- 4}$

Step 2.3

Simplify the right side.

Tap for more steps...

Step 2.3.1

Simplify each term.

Tap for more steps...

Step 2.3.1.1

Divide $36$ by $- 4$ .

$y^{2} = - 9 + \frac{- 9 x^{2}}{- 4} + \frac{- 36 z^{2}}{- 4}$

Step 2.3.1.2

Dividing two negative values results in a positive value.

$y^{2} = - 9 + \frac{9 x^{2}}{4} + \frac{- 36 z^{2}}{- 4}$

Step 2.3.1.3

Cancel the common factor of $- 36$ and $- 4$ .

Tap for more steps...

Step 2.3.1.3.1

Factor $- 4$ out of $- 36 z^{2}$ .

$y^{2} = - 9 + \frac{9 x^{2}}{4} + \frac{- 4 (9 z^{2})}{- 4}$

Step 2.3.1.3.2

Cancel the common factors.

Tap for more steps...

Step 2.3.1.3.2.1

Factor $- 4$ out of $- 4$ .

$y^{2} = - 9 + \frac{9 x^{2}}{4} + \frac{- 4 (9 z^{2})}{- 4 (1)}$

Step 2.3.1.3.2.2

Cancel the common factor.

$y^{2} = - 9 + \frac{9 x^{2}}{4} + \frac{- 4 (9 z^{2})}{- 4 \cdot 1}$

Step 2.3.1.3.2.3

Rewrite the expression.

$y^{2} = - 9 + \frac{9 x^{2}}{4} + \frac{9 z^{2}}{1}$

Step 2.3.1.3.2.4

Divide $9 z^{2}$ by $1$ .

$y^{2} = - 9 + \frac{9 x^{2}}{4} + 9 z^{2}$

Step 3

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

$y = \pm \sqrt{- 9 + \frac{9 x^{2}}{4} + 9 z^{2}}$

Step 4

Simplify $\pm \sqrt{- 9 + \frac{9 x^{2}}{4} + 9 z^{2}}$ .

Tap for more steps...

Step 4.1

Factor $9$ out of $- 9 + \frac{9 x^{2}}{4} + 9 z^{2}$ .

Tap for more steps...

Step 4.1.1

Factor $9$ out of $- 9$ .

$y = \pm \sqrt{9 \cdot - 1 + \frac{9 x^{2}}{4} + 9 z^{2}}$

Step 4.1.2

Factor $9$ out of $\frac{9 x^{2}}{4}$ .

$y = \pm \sqrt{9 \cdot - 1 + 9 \frac{x^{2}}{4} + 9 z^{2}}$

Step 4.1.3

Factor $9$ out of $9 \cdot - 1 + 9 \frac{x^{2}}{4}$ .

$y = \pm \sqrt{9 \cdot (- 1 + \frac{x^{2}}{4}) + 9 z^{2}}$

Step 4.1.4

Factor $9$ out of $9 \cdot (- 1 + \frac{x^{2}}{4}) + 9 z^{2}$ .

$y = \pm \sqrt{9 (- 1 + \frac{x^{2}}{4} + z^{2})}$

Step 4.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$y = \pm \sqrt{9 (- 1 \cdot \frac{4}{4} + \frac{x^{2}}{4} + z^{2})}$

Step 4.3

Simplify terms.

Tap for more steps...

Step 4.3.1

Combine $- 1$ and $\frac{4}{4}$ .

$y = \pm \sqrt{9 (\frac{- 1 \cdot 4}{4} + \frac{x^{2}}{4} + z^{2})}$

Step 4.3.2

Combine the numerators over the common denominator.

$y = \pm \sqrt{9 (\frac{- 1 \cdot 4 + x^{2}}{4} + z^{2})}$

Step 4.3.3

Multiply $- 1$ by $4$ .

$y = \pm \sqrt{9 (\frac{- 4 + x^{2}}{4} + z^{2})}$

Step 4.4

Simplify the numerator.

Tap for more steps...

Step 4.4.1

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

$y = \pm \sqrt{9 (\frac{- 2^{2} + x^{2}}{4} + z^{2})}$

Step 4.4.2

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

$y = \pm \sqrt{9 (\frac{x^{2} - 2^{2}}{4} + z^{2})}$

Step 4.4.3

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

$y = \pm \sqrt{9 (\frac{(x + 2) (x - 2)}{4} + z^{2})}$

Step 4.5

To write $z^{2}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$y = \pm \sqrt{9 (\frac{(x + 2) (x - 2)}{4} + z^{2} \cdot \frac{4}{4})}$

Step 4.6

Simplify terms.

Tap for more steps...

Step 4.6.1

Combine $z^{2}$ and $\frac{4}{4}$ .

$y = \pm \sqrt{9 (\frac{(x + 2) (x - 2)}{4} + \frac{z^{2} \cdot 4}{4})}$

Step 4.6.2

Combine the numerators over the common denominator.

$y = \pm \sqrt{9 \frac{(x + 2) (x - 2) + z^{2} \cdot 4}{4}}$

Step 4.7

Simplify the numerator.

Tap for more steps...

Step 4.7.1

Expand $(x + 2) (x - 2)$ using the FOIL Method.

Tap for more steps...

Step 4.7.1.1

Apply the distributive property.

$y = \pm \sqrt{9 \frac{x (x - 2) + 2 (x - 2) + z^{2} \cdot 4}{4}}$

Step 4.7.1.2

Apply the distributive property.

$y = \pm \sqrt{9 \frac{x \cdot x + x \cdot - 2 + 2 (x - 2) + z^{2} \cdot 4}{4}}$

Step 4.7.1.3

Apply the distributive property.

$y = \pm \sqrt{9 \frac{x \cdot x + x \cdot - 2 + 2 x + 2 \cdot - 2 + z^{2} \cdot 4}{4}}$

Step 4.7.2

Combine the opposite terms in $x \cdot x + x \cdot - 2 + 2 x + 2 \cdot - 2$ .

Tap for more steps...

Step 4.7.2.1

Reorder the factors in the terms $x \cdot - 2$ and $2 x$ .

$y = \pm \sqrt{9 \frac{x \cdot x - 2 x + 2 x + 2 \cdot - 2 + z^{2} \cdot 4}{4}}$

Step 4.7.2.2

Add $- 2 x$ and $2 x$ .

$y = \pm \sqrt{9 \frac{x \cdot x + 0 + 2 \cdot - 2 + z^{2} \cdot 4}{4}}$

Step 4.7.2.3

Add $x \cdot x$ and $0$ .

$y = \pm \sqrt{9 \frac{x \cdot x + 2 \cdot - 2 + z^{2} \cdot 4}{4}}$

Step 4.7.3

Simplify each term.

Tap for more steps...

Step 4.7.3.1

Multiply $x$ by $x$ .

$y = \pm \sqrt{9 \frac{x^{2} + 2 \cdot - 2 + z^{2} \cdot 4}{4}}$

Step 4.7.3.2

Multiply $2$ by $- 2$ .

$y = \pm \sqrt{9 \frac{x^{2} - 4 + z^{2} \cdot 4}{4}}$

Step 4.7.4

Move $4$ to the left of $z^{2}$ .

$y = \pm \sqrt{9 \frac{x^{2} - 4 + 4 z^{2}}{4}}$

Step 4.8

Combine $9$ and $\frac{x^{2} - 4 + 4 z^{2}}{4}$ .

$y = \pm \sqrt{\frac{9 (x^{2} - 4 + 4 z^{2})}{4}}$

Step 4.9

Rewrite $\frac{9 (x^{2} - 4 + 4 z^{2})}{4}$ as ${(\frac{3}{2})}^{2} (x^{2} - 4 + {(2 z)}^{2})$ .

Tap for more steps...

Step 4.9.1

Factor the perfect power $3^{2}$ out of $9 (x^{2} - 4 + 4 z^{2})$ .

$y = \pm \sqrt{\frac{3^{2} (x^{2} - 4 + {(2 z)}^{2})}{4}}$

Step 4.9.2

Factor the perfect power $2^{2}$ out of $4$ .

$y = \pm \sqrt{\frac{3^{2} (x^{2} - 4 + {(2 z)}^{2})}{2^{2} \cdot 1}}$

Step 4.9.3

Rearrange the fraction $\frac{3^{2} (x^{2} - 4 + {(2 z)}^{2})}{2^{2} \cdot 1}$ .

$y = \pm \sqrt{{(\frac{3}{2})}^{2} (x^{2} - 4 + {(2 z)}^{2})}$

Step 4.10

Pull terms out from under the radical.

$y = \pm \frac{3}{2} \sqrt{x^{2} - 4 + {(2 z)}^{2}}$

Step 4.11

Simplify the expression.

Tap for more steps...

Step 4.11.1

Apply the product rule to $2 z$ .

$y = \pm \frac{3}{2} \sqrt{x^{2} - 4 + 2^{2} z^{2}}$

Step 4.11.2

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

$y = \pm \frac{3}{2} \sqrt{x^{2} - 4 + 4 z^{2}}$

Step 4.12

Combine $\frac{3}{2}$ and $\sqrt{x^{2} - 4 + 4 z^{2}}$ .

$y = \pm \frac{3 \sqrt{x^{2} - 4 + 4 z^{2}}}{2}$

Step 5

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

Tap for more steps...

Step 5.1

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

$y = \frac{3 \sqrt{x^{2} - 4 + 4 z^{2}}}{2}$

Step 5.2

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

$y = - \frac{3 \sqrt{x^{2} - 4 + 4 z^{2}}}{2}$

Step 5.3

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

$y = \frac{3 \sqrt{x^{2} - 4 + 4 z^{2}}}{2}$

$y = - \frac{3 \sqrt{x^{2} - 4 + 4 z^{2}}}{2}$

$y = \frac{3 \sqrt{x^{2} - 4 + 4 z^{2}}}{2}$

$y = - \frac{3 \sqrt{x^{2} - 4 + 4 z^{2}}}{2}$

Step 6

Set the radicand in $\sqrt{x^{2} - 4 + 4 z^{2}}$ greater than or equal to $0$ to find where the expression is defined.

$x^{2} - 4 + 4 z^{2} \geq 0$

Step 7

Solve for $x$ .

Tap for more steps...

Step 7.1

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

Tap for more steps...

Step 7.1.1

Add $4$ to both sides of the inequality.

$x^{2} + 4 z^{2} \geq 4$

Step 7.1.2

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

$x^{2} \geq 4 - 4 z^{2}$

Step 7.2

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

$\sqrt{x^{2}} \geq \sqrt{4 - 4 z^{2}}$

Step 7.3

Simplify the equation.

Tap for more steps...

Step 7.3.1

Simplify the left side.

Tap for more steps...

Step 7.3.1.1

Pull terms out from under the radical.

$| x | \geq \sqrt{4 - 4 z^{2}}$

Step 7.3.2

Simplify the right side.

Tap for more steps...

Step 7.3.2.1

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

Tap for more steps...

Step 7.3.2.1.1

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

Tap for more steps...

Step 7.3.2.1.1.1

Factor $4$ out of $4$ .

$| x | \geq \sqrt{4 (1) - 4 z^{2}}$

Step 7.3.2.1.1.2

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

$| x | \geq \sqrt{4 (1) + 4 (- z^{2})}$

Step 7.3.2.1.1.3

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

$| x | \geq \sqrt{4 (1 - z^{2})}$

Step 7.3.2.1.2

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

$| x | \geq \sqrt{4 (1^{2} - z^{2})}$

Step 7.3.2.1.3

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 $b = z$ .

$| x | \geq \sqrt{4 (1 + z) (1 - z)}$

Step 7.3.2.1.4

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

Tap for more steps...

Step 7.3.2.1.4.1

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

$| x | \geq \sqrt{2^{2} (1 + z) (1 - z)}$

Step 7.3.2.1.4.2

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

$| x | \geq \sqrt{2^{2} (1^{2} + z) (1 - z)}$

Step 7.3.2.1.4.3

Add parentheses.

$| x | \geq \sqrt{2^{2} ((1^{2} + z) (1 - z))}$

Step 7.3.2.1.5

Pull terms out from under the radical.

$| x | \geq | 2 | \sqrt{(1^{2} + z) (1 - z)}$

Step 7.3.2.1.6

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

$| x | \geq 2 \sqrt{(1^{2} + z) (1 - z)}$

Step 7.3.2.1.7

One to any power is one.

$| x | \geq 2 \sqrt{(1 + z) (1 - z)}$

Step 7.4

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

Tap for more steps...

Step 7.4.1

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

$x \geq 0$

Step 7.4.2

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

$x \geq 2 \sqrt{(1 + z) (1 - z)}$

Step 7.4.3

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

Tap for more steps...

Step 7.4.3.1

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

Tap for more steps...

Step 7.4.3.1.1

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

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

Step 7.4.3.1.2

Solve for $x$ .

Tap for more steps...

Step 7.4.3.1.2.1

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

Tap for more steps...

Step 7.4.3.1.2.1.1

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

Tap for more steps...

Step 7.4.3.1.2.1.1.1

Apply the distributive property.

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

Step 7.4.3.1.2.1.1.2

Apply the distributive property.

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

Step 7.4.3.1.2.1.1.3

Apply the distributive property.

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

Step 7.4.3.1.2.1.2

Simplify and combine like terms.

Tap for more steps...

Step 7.4.3.1.2.1.2.1

Simplify each term.

Tap for more steps...

Step 7.4.3.1.2.1.2.1.1

Multiply $1$ by $1$ .

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

Step 7.4.3.1.2.1.2.1.2

Multiply $- z$ by $1$ .

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

Step 7.4.3.1.2.1.2.1.3

Multiply $z$ by $1$ .

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

Step 7.4.3.1.2.1.2.1.4

Rewrite using the commutative property of multiplication.

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

Step 7.4.3.1.2.1.2.1.5

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

Tap for more steps...

Step 7.4.3.1.2.1.2.1.5.1

Move $z$ .

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

Step 7.4.3.1.2.1.2.1.5.2

Multiply $z$ by $z$ .

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

Step 7.4.3.1.2.1.2.2

Add $- z$ and $z$ .

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

Step 7.4.3.1.2.1.2.3

Add $1$ and $0$ .

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

Step 7.4.3.1.2.2

Subtract $1$ from both sides of the inequality.

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

Step 7.4.3.1.2.3

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

Tap for more steps...

Step 7.4.3.1.2.3.1

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

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

Step 7.4.3.1.2.3.2

Simplify the left side.

Tap for more steps...

Step 7.4.3.1.2.3.2.1

Dividing two negative values results in a positive value.

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

Step 7.4.3.1.2.3.2.2

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

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

Step 7.4.3.1.2.3.3

Simplify the right side.

Tap for more steps...

Step 7.4.3.1.2.3.3.1

Divide $- 1$ by $- 1$ .

$z^{2} \leq 1$

Step 7.4.3.1.2.4

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

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

Step 7.4.3.1.2.5

Simplify the equation.

Tap for more steps...

Step 7.4.3.1.2.5.1

Simplify the left side.

Tap for more steps...

Step 7.4.3.1.2.5.1.1

Pull terms out from under the radical.

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

Step 7.4.3.1.2.5.2

Simplify the right side.

Tap for more steps...

Step 7.4.3.1.2.5.2.1

Any root of $1$ is $1$ .

$| z | \leq 1$

Step 7.4.3.1.2.6

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

Tap for more steps...

Step 7.4.3.1.2.6.1

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

$z \geq 0$

Step 7.4.3.1.2.6.2

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

$z \leq 1$

Step 7.4.3.1.2.6.3

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

$z < 0$

Step 7.4.3.1.2.6.4

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

$- z \leq 1$

Step 7.4.3.1.2.6.5

Write as a piecewise.

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

Step 7.4.3.1.2.7

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

$0 \leq z \leq 1$

Step 7.4.3.1.2.8

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

Tap for more steps...

Step 7.4.3.1.2.8.1

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

Tap for more steps...

Step 7.4.3.1.2.8.1.1

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

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

Step 7.4.3.1.2.8.1.2

Simplify the left side.

Tap for more steps...

Step 7.4.3.1.2.8.1.2.1

Dividing two negative values results in a positive value.

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

Step 7.4.3.1.2.8.1.2.2

Divide $z$ by $1$ .

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

Step 7.4.3.1.2.8.1.3

Simplify the right side.

Tap for more steps...

Step 7.4.3.1.2.8.1.3.1

Divide $1$ by $- 1$ .

$z \geq - 1$

Step 7.4.3.1.2.8.2

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

$- 1 \leq z < 0$

Step 7.4.3.1.2.9

Find the union of the solutions.

$- 1 \leq z \leq 1$

Step 7.4.3.1.3

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

$[- 1, 1]$

Step 7.4.3.2

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

$0 \leq x \leq 1$

Step 7.4.4

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

$x < 0$

Step 7.4.5

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

$- x \geq 2 \sqrt{(1 + z) (1 - z)}$

Step 7.4.6

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

Tap for more steps...

Step 7.4.6.1

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

Tap for more steps...

Step 7.4.6.1.1

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

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

Step 7.4.6.1.2

Solve for $x$ .

Tap for more steps...

Step 7.4.6.1.2.1

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

Tap for more steps...

Step 7.4.6.1.2.1.1

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

Tap for more steps...

Step 7.4.6.1.2.1.1.1

Apply the distributive property.

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

Step 7.4.6.1.2.1.1.2

Apply the distributive property.

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

Step 7.4.6.1.2.1.1.3

Apply the distributive property.

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

Step 7.4.6.1.2.1.2

Simplify and combine like terms.

Tap for more steps...

Step 7.4.6.1.2.1.2.1

Simplify each term.

Tap for more steps...

Step 7.4.6.1.2.1.2.1.1

Multiply $1$ by $1$ .

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

Step 7.4.6.1.2.1.2.1.2

Multiply $- z$ by $1$ .

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

Step 7.4.6.1.2.1.2.1.3

Multiply $z$ by $1$ .

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

Step 7.4.6.1.2.1.2.1.4

Rewrite using the commutative property of multiplication.

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

Step 7.4.6.1.2.1.2.1.5

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

Tap for more steps...

Step 7.4.6.1.2.1.2.1.5.1

Move $z$ .

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

Step 7.4.6.1.2.1.2.1.5.2

Multiply $z$ by $z$ .

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

Step 7.4.6.1.2.1.2.2

Add $- z$ and $z$ .

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

Step 7.4.6.1.2.1.2.3

Add $1$ and $0$ .

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

Step 7.4.6.1.2.2

Subtract $1$ from both sides of the inequality.

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

Step 7.4.6.1.2.3

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

Tap for more steps...

Step 7.4.6.1.2.3.1

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

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

Step 7.4.6.1.2.3.2

Simplify the left side.

Tap for more steps...

Step 7.4.6.1.2.3.2.1

Dividing two negative values results in a positive value.

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

Step 7.4.6.1.2.3.2.2

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

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

Step 7.4.6.1.2.3.3

Simplify the right side.

Tap for more steps...

Step 7.4.6.1.2.3.3.1

Divide $- 1$ by $- 1$ .

$z^{2} \leq 1$

Step 7.4.6.1.2.4

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

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

Step 7.4.6.1.2.5

Simplify the equation.

Tap for more steps...

Step 7.4.6.1.2.5.1

Simplify the left side.

Tap for more steps...

Step 7.4.6.1.2.5.1.1

Pull terms out from under the radical.

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

Step 7.4.6.1.2.5.2

Simplify the right side.

Tap for more steps...

Step 7.4.6.1.2.5.2.1

Any root of $1$ is $1$ .

$| z | \leq 1$

Step 7.4.6.1.2.6

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

Tap for more steps...

Step 7.4.6.1.2.6.1

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

$z \geq 0$

Step 7.4.6.1.2.6.2

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

$z \leq 1$

Step 7.4.6.1.2.6.3

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

$z < 0$

Step 7.4.6.1.2.6.4

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

$- z \leq 1$

Step 7.4.6.1.2.6.5

Write as a piecewise.

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

Step 7.4.6.1.2.7

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

$0 \leq z \leq 1$

Step 7.4.6.1.2.8

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

Tap for more steps...

Step 7.4.6.1.2.8.1

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

Tap for more steps...

Step 7.4.6.1.2.8.1.1

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

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

Step 7.4.6.1.2.8.1.2

Simplify the left side.

Tap for more steps...

Step 7.4.6.1.2.8.1.2.1

Dividing two negative values results in a positive value.

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

Step 7.4.6.1.2.8.1.2.2

Divide $z$ by $1$ .

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

Step 7.4.6.1.2.8.1.3

Simplify the right side.

Tap for more steps...

Step 7.4.6.1.2.8.1.3.1

Divide $1$ by $- 1$ .

$z \geq - 1$

Step 7.4.6.1.2.8.2

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

$- 1 \leq z < 0$

Step 7.4.6.1.2.9

Find the union of the solutions.

$- 1 \leq z \leq 1$

Step 7.4.6.1.3

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

$[- 1, 1]$

Step 7.4.6.2

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

$- 1 \leq x < 0$

Step 7.4.7

Write as a piecewise.

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

Step 7.5

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

No solution

Step 7.6

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

Tap for more steps...

Step 7.6.1

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

Tap for more steps...

Step 7.6.1.1

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

$\frac{- x}{- 1} \leq \frac{2 \sqrt{(1 + z) (1 - z)}}{- 1}$

Step 7.6.1.2

Simplify the left side.

Tap for more steps...

Step 7.6.1.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} \leq \frac{2 \sqrt{(1 + z) (1 - z)}}{- 1}$

Step 7.6.1.2.2

Divide $x$ by $1$ .

$x \leq \frac{2 \sqrt{(1 + z) (1 - z)}}{- 1}$

Step 7.6.1.3

Simplify the right side.

Tap for more steps...

Step 7.6.1.3.1

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

$x \leq - 1 \cdot (2 \sqrt{(1 + z) (1 - z)})$

Step 7.6.1.3.2

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

$x \leq - (2 \sqrt{(1 + z) (1 - z)})$

Step 7.6.1.3.3

Multiply $2$ by $- 1$ .

$x \leq - 2 \sqrt{(1 + z) (1 - z)}$

Step 7.6.2

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

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

Step 7.7

Find the union of the solutions.

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

Step 8

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