Pre-Algebra Examples

$9 x^{2} + 16 y^{2} > 144$

Step 1

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

$16 y^{2} > 144 - 9 x^{2}$

Step 2

Divide each term in $16 y^{2} > 144 - 9 x^{2}$ by $16$ and simplify.

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Step 2.1

Divide each term in $16 y^{2} > 144 - 9 x^{2}$ by $16$ .

$\frac{16 y^{2}}{16} > \frac{144}{16} + \frac{- 9 x^{2}}{16}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $16$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{16 y^{2}}{16} > \frac{144}{16} + \frac{- 9 x^{2}}{16}$

Step 2.2.1.2

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

$y^{2} > \frac{144}{16} + \frac{- 9 x^{2}}{16}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

Divide $144$ by $16$ .

$y^{2} > 9 + \frac{- 9 x^{2}}{16}$

Step 2.3.1.2

Move the negative in front of the fraction.

$y^{2} > 9 - \frac{9 x^{2}}{16}$

Step 3

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

$\sqrt{y^{2}} > \sqrt{9 - \frac{9 x^{2}}{16}}$

Step 4

Simplify the equation.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Pull terms out from under the radical.

$| y | > \sqrt{9 - \frac{9 x^{2}}{16}}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $\sqrt{9 - \frac{9 x^{2}}{16}}$ .

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Step 4.2.1.1

Write the expression using exponents.

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Step 4.2.1.1.1

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

$| y | > \sqrt{3^{2} - \frac{9 x^{2}}{16}}$

Step 4.2.1.1.2

Rewrite $\frac{9 x^{2}}{16}$ as ${(\frac{3 x}{4})}^{2}$ .

$| y | > \sqrt{3^{2} - {(\frac{3 x}{4})}^{2}}$

Step 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 $a = 3$ and $b = \frac{3 x}{4}$ .

$| y | > \sqrt{(3 + \frac{3 x}{4}) (3 - \frac{3 x}{4})}$

Step 4.2.1.3

To write $3$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$| y | > \sqrt{(3 \cdot \frac{4}{4} + \frac{3 x}{4}) (3 - \frac{3 x}{4})}$

Step 4.2.1.4

Combine $3$ and $\frac{4}{4}$ .

$| y | > \sqrt{(\frac{3 \cdot 4}{4} + \frac{3 x}{4}) (3 - \frac{3 x}{4})}$

Step 4.2.1.5

Combine the numerators over the common denominator.

$| y | > \sqrt{\frac{3 \cdot 4 + 3 x}{4} (3 - \frac{3 x}{4})}$

Step 4.2.1.6

Factor $3$ out of $3 \cdot 4 + 3 x$ .

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Step 4.2.1.6.1

Factor $3$ out of $3 \cdot 4$ .

$| y | > \sqrt{\frac{3 (4) + 3 x}{4} (3 - \frac{3 x}{4})}$

Step 4.2.1.6.2

Factor $3$ out of $3 x$ .

$| y | > \sqrt{\frac{3 (4) + 3 (x)}{4} (3 - \frac{3 x}{4})}$

Step 4.2.1.6.3

Factor $3$ out of $3 (4) + 3 (x)$ .

$| y | > \sqrt{\frac{3 (4 + x)}{4} (3 - \frac{3 x}{4})}$

Step 4.2.1.7

To write $3$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$| y | > \sqrt{\frac{3 (4 + x)}{4} (3 \cdot \frac{4}{4} - \frac{3 x}{4})}$

Step 4.2.1.8

Combine $3$ and $\frac{4}{4}$ .

$| y | > \sqrt{\frac{3 (4 + x)}{4} (\frac{3 \cdot 4}{4} - \frac{3 x}{4})}$

Step 4.2.1.9

Combine the numerators over the common denominator.

$| y | > \sqrt{\frac{3 (4 + x)}{4} \cdot \frac{3 \cdot 4 - 3 x}{4}}$

Step 4.2.1.10

Factor $3$ out of $3 \cdot 4 - 3 x$ .

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Step 4.2.1.10.1

Factor $3$ out of $3 \cdot 4$ .

$| y | > \sqrt{\frac{3 (4 + x)}{4} \cdot \frac{3 (4) - 3 x}{4}}$

Step 4.2.1.10.2

Factor $3$ out of $- 3 x$ .

$| y | > \sqrt{\frac{3 (4 + x)}{4} \cdot \frac{3 (4) + 3 (- x)}{4}}$

Step 4.2.1.10.3

Factor $3$ out of $3 (4) + 3 (- x)$ .

$| y | > \sqrt{\frac{3 (4 + x)}{4} \cdot \frac{3 (4 - x)}{4}}$

Step 4.2.1.11

Multiply $\frac{3 (4 + x)}{4}$ by $\frac{3 (4 - x)}{4}$ .

$| y | > \sqrt{\frac{3 (4 + x) (3 (4 - x))}{4 \cdot 4}}$

Step 4.2.1.12

Multiply.

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Step 4.2.1.12.1

Multiply $3$ by $3$ .

$| y | > \sqrt{\frac{9 (4 + x) (4 - x)}{4 \cdot 4}}$

Step 4.2.1.12.2

Multiply $4$ by $4$ .

$| y | > \sqrt{\frac{9 (4 + x) (4 - x)}{16}}$

Step 4.2.1.13

Rewrite $\frac{9 (4 + x) (4 - x)}{16}$ as ${(\frac{3}{4})}^{2} ((2^{2} + x) (4 - x))$ .

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Step 4.2.1.13.1

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

$| y | > \sqrt{\frac{3^{2} ((2^{2} + x) (4 - x))}{16}}$

Step 4.2.1.13.2

Factor the perfect power $4^{2}$ out of $16$ .

$| y | > \sqrt{\frac{3^{2} ((2^{2} + x) (4 - x))}{4^{2} \cdot 1}}$

Step 4.2.1.13.3

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

$| y | > \sqrt{{(\frac{3}{4})}^{2} ((2^{2} + x) (4 - x))}$

Step 4.2.1.14

Pull terms out from under the radical.

$| y | > | \frac{3}{4} | \sqrt{(2^{2} + x) (4 - x)}$

Step 4.2.1.15

$\frac{3}{4}$ is approximately $0.75$ which is positive so remove the absolute value

$| y | > \frac{3}{4} \sqrt{(2^{2} + x) (4 - x)}$

Step 4.2.1.16

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

$| y | > \frac{3}{4} \sqrt{(4 + x) (4 - x)}$

Step 4.2.1.17

Combine $\frac{3}{4}$ and $\sqrt{(4 + x) (4 - x)}$ .

$| y | > \frac{3 \sqrt{(4 + x) (4 - x)}}{4}$

Step 5

Write $| y | > \frac{3 \sqrt{(4 + x) (4 - x)}}{4}$ as a piecewise.

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Step 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 5.2

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

$y > \frac{3 \sqrt{(4 + x) (4 - x)}}{4}$

Step 5.3

Find the domain of $y > \frac{3 \sqrt{(4 + x) (4 - x)}}{4}$ and find the intersection with $y \geq 0$ .

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Step 5.3.1

Find the domain of $y > \frac{3 \sqrt{(4 + x) (4 - x)}}{4}$ .

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Step 5.3.1.1

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

$(4 + x) (4 - x) \geq 0$

Step 5.3.1.2

Solve for $y$ .

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Step 5.3.1.2.1

Simplify $(4 + x) (4 - x)$ .

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Step 5.3.1.2.1.1

Expand $(4 + x) (4 - x)$ using the FOIL Method.

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Step 5.3.1.2.1.1.1

Apply the distributive property.

$4 (4 - x) + x (4 - x) \geq 0$

Step 5.3.1.2.1.1.2

Apply the distributive property.

$4 \cdot 4 + 4 (- x) + x (4 - x) \geq 0$

Step 5.3.1.2.1.1.3

Apply the distributive property.

$4 \cdot 4 + 4 (- x) + x \cdot 4 + x (- x) \geq 0$

Step 5.3.1.2.1.2

Simplify and combine like terms.

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Step 5.3.1.2.1.2.1

Simplify each term.

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Step 5.3.1.2.1.2.1.1

Multiply $4$ by $4$ .

$16 + 4 (- x) + x \cdot 4 + x (- x) \geq 0$

Step 5.3.1.2.1.2.1.2

Multiply $- 1$ by $4$ .

$16 - 4 x + x \cdot 4 + x (- x) \geq 0$

Step 5.3.1.2.1.2.1.3

Move $4$ to the left of $x$ .

$16 - 4 x + 4 \cdot x + x (- x) \geq 0$

Step 5.3.1.2.1.2.1.4

Rewrite using the commutative property of multiplication.

$16 - 4 x + 4 x - x \cdot x \geq 0$

Step 5.3.1.2.1.2.1.5

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

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Step 5.3.1.2.1.2.1.5.1

Move $x$ .

$16 - 4 x + 4 x - (x \cdot x) \geq 0$

Step 5.3.1.2.1.2.1.5.2

Multiply $x$ by $x$ .

$16 - 4 x + 4 x - x^{2} \geq 0$

Step 5.3.1.2.1.2.2

Add $- 4 x$ and $4 x$ .

$16 + 0 - x^{2} \geq 0$

Step 5.3.1.2.1.2.3

Add $16$ and $0$ .

$16 - x^{2} \geq 0$

Step 5.3.1.2.2

Subtract $16$ from both sides of the inequality.

$- x^{2} \geq - 16$

Step 5.3.1.2.3

Divide each term in $- x^{2} \geq - 16$ by $- 1$ and simplify.

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Step 5.3.1.2.3.1

Divide each term in $- x^{2} \geq - 16$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x^{2}}{- 1} \leq \frac{- 16}{- 1}$

Step 5.3.1.2.3.2

Simplify the left side.

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Step 5.3.1.2.3.2.1

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} \leq \frac{- 16}{- 1}$

Step 5.3.1.2.3.2.2

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

$x^{2} \leq \frac{- 16}{- 1}$

Step 5.3.1.2.3.3

Simplify the right side.

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Step 5.3.1.2.3.3.1

Divide $- 16$ by $- 1$ .

$x^{2} \leq 16$

Step 5.3.1.2.4

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

$\sqrt{x^{2}} \leq \sqrt{16}$

Step 5.3.1.2.5

Simplify the equation.

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Step 5.3.1.2.5.1

Simplify the left side.

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Step 5.3.1.2.5.1.1

Pull terms out from under the radical.

$| x | \leq \sqrt{16}$

Step 5.3.1.2.5.2

Simplify the right side.

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Step 5.3.1.2.5.2.1

Simplify $\sqrt{16}$ .

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Step 5.3.1.2.5.2.1.1

Rewrite $16$ as $4^{2}$ .

$| x | \leq \sqrt{4^{2}}$

Step 5.3.1.2.5.2.1.2

Pull terms out from under the radical.

$| x | \leq | 4 |$

Step 5.3.1.2.5.2.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $4$ is $4$ .

$| x | \leq 4$

Step 5.3.1.2.6

Write $| x | \leq 4$ as a piecewise.

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Step 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 5.3.1.2.6.2

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

$x \leq 4$

Step 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 5.3.1.2.6.4

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

$- x \leq 4$

Step 5.3.1.2.6.5

Write as a piecewise.

${\begin{cases} x \leq 4 & x \geq 0 \\ - x \leq 4 & x < 0 \end{cases}$

Step 5.3.1.2.7

Find the intersection of $x \leq 4$ and $x \geq 0$ .

$0 \leq x \leq 4$

Step 5.3.1.2.8

Solve $- x \leq 4$ when $x < 0$ .

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Step 5.3.1.2.8.1

Divide each term in $- x \leq 4$ by $- 1$ and simplify.

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Step 5.3.1.2.8.1.1

Divide each term in $- x \leq 4$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \geq \frac{4}{- 1}$

Step 5.3.1.2.8.1.2

Simplify the left side.

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Step 5.3.1.2.8.1.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} \geq \frac{4}{- 1}$

Step 5.3.1.2.8.1.2.2

Divide $x$ by $1$ .

$x \geq \frac{4}{- 1}$

Step 5.3.1.2.8.1.3

Simplify the right side.

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Step 5.3.1.2.8.1.3.1

Divide $4$ by $- 1$ .

$x \geq - 4$

Step 5.3.1.2.8.2

Find the intersection of $x \geq - 4$ and $x < 0$ .

$- 4 \leq x < 0$

Step 5.3.1.2.9

Find the union of the solutions.

$- 4 \leq x \leq 4$

Step 5.3.1.3

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

$[- 4, 4]$

Step 5.3.2

Find the intersection of $y \geq 0$ and $[- 4, 4]$ .

$0 \leq y \leq 4$

Step 5.4

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

$y < 0$

Step 5.5

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

$- y > \frac{3 \sqrt{(4 + x) (4 - x)}}{4}$

Step 5.6

Find the domain of $- y > \frac{3 \sqrt{(4 + x) (4 - x)}}{4}$ and find the intersection with $y < 0$ .

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Step 5.6.1

Find the domain of $- y > \frac{3 \sqrt{(4 + x) (4 - x)}}{4}$ .

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Step 5.6.1.1

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

$(4 + x) (4 - x) \geq 0$

Step 5.6.1.2

Solve for $y$ .

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Step 5.6.1.2.1

Simplify $(4 + x) (4 - x)$ .

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Step 5.6.1.2.1.1

Expand $(4 + x) (4 - x)$ using the FOIL Method.

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Step 5.6.1.2.1.1.1

Apply the distributive property.

$4 (4 - x) + x (4 - x) \geq 0$

Step 5.6.1.2.1.1.2

Apply the distributive property.

$4 \cdot 4 + 4 (- x) + x (4 - x) \geq 0$

Step 5.6.1.2.1.1.3

Apply the distributive property.

$4 \cdot 4 + 4 (- x) + x \cdot 4 + x (- x) \geq 0$

Step 5.6.1.2.1.2

Simplify and combine like terms.

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Step 5.6.1.2.1.2.1

Simplify each term.

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Step 5.6.1.2.1.2.1.1

Multiply $4$ by $4$ .

$16 + 4 (- x) + x \cdot 4 + x (- x) \geq 0$

Step 5.6.1.2.1.2.1.2

Multiply $- 1$ by $4$ .

$16 - 4 x + x \cdot 4 + x (- x) \geq 0$

Step 5.6.1.2.1.2.1.3

Move $4$ to the left of $x$ .

$16 - 4 x + 4 \cdot x + x (- x) \geq 0$

Step 5.6.1.2.1.2.1.4

Rewrite using the commutative property of multiplication.

$16 - 4 x + 4 x - x \cdot x \geq 0$

Step 5.6.1.2.1.2.1.5

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

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Step 5.6.1.2.1.2.1.5.1

Move $x$ .

$16 - 4 x + 4 x - (x \cdot x) \geq 0$

Step 5.6.1.2.1.2.1.5.2

Multiply $x$ by $x$ .

$16 - 4 x + 4 x - x^{2} \geq 0$

Step 5.6.1.2.1.2.2

Add $- 4 x$ and $4 x$ .

$16 + 0 - x^{2} \geq 0$

Step 5.6.1.2.1.2.3

Add $16$ and $0$ .

$16 - x^{2} \geq 0$

Step 5.6.1.2.2

Subtract $16$ from both sides of the inequality.

$- x^{2} \geq - 16$

Step 5.6.1.2.3

Divide each term in $- x^{2} \geq - 16$ by $- 1$ and simplify.

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Step 5.6.1.2.3.1

Divide each term in $- x^{2} \geq - 16$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x^{2}}{- 1} \leq \frac{- 16}{- 1}$

Step 5.6.1.2.3.2

Simplify the left side.

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Step 5.6.1.2.3.2.1

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} \leq \frac{- 16}{- 1}$

Step 5.6.1.2.3.2.2

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

$x^{2} \leq \frac{- 16}{- 1}$

Step 5.6.1.2.3.3

Simplify the right side.

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Step 5.6.1.2.3.3.1

Divide $- 16$ by $- 1$ .

$x^{2} \leq 16$

Step 5.6.1.2.4

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

$\sqrt{x^{2}} \leq \sqrt{16}$

Step 5.6.1.2.5

Simplify the equation.

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Step 5.6.1.2.5.1

Simplify the left side.

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Step 5.6.1.2.5.1.1

Pull terms out from under the radical.

$| x | \leq \sqrt{16}$

Step 5.6.1.2.5.2

Simplify the right side.

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Step 5.6.1.2.5.2.1

Simplify $\sqrt{16}$ .

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Step 5.6.1.2.5.2.1.1

Rewrite $16$ as $4^{2}$ .

$| x | \leq \sqrt{4^{2}}$

Step 5.6.1.2.5.2.1.2

Pull terms out from under the radical.

$| x | \leq | 4 |$

Step 5.6.1.2.5.2.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $4$ is $4$ .

$| x | \leq 4$

Step 5.6.1.2.6

Write $| x | \leq 4$ as a piecewise.

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Step 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 5.6.1.2.6.2

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

$x \leq 4$

Step 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 5.6.1.2.6.4

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

$- x \leq 4$

Step 5.6.1.2.6.5

Write as a piecewise.

${\begin{cases} x \leq 4 & x \geq 0 \\ - x \leq 4 & x < 0 \end{cases}$

Step 5.6.1.2.7

Find the intersection of $x \leq 4$ and $x \geq 0$ .

$0 \leq x \leq 4$

Step 5.6.1.2.8

Solve $- x \leq 4$ when $x < 0$ .

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Step 5.6.1.2.8.1

Divide each term in $- x \leq 4$ by $- 1$ and simplify.

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Step 5.6.1.2.8.1.1

Divide each term in $- x \leq 4$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \geq \frac{4}{- 1}$

Step 5.6.1.2.8.1.2

Simplify the left side.

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Step 5.6.1.2.8.1.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} \geq \frac{4}{- 1}$

Step 5.6.1.2.8.1.2.2

Divide $x$ by $1$ .

$x \geq \frac{4}{- 1}$

Step 5.6.1.2.8.1.3

Simplify the right side.

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Step 5.6.1.2.8.1.3.1

Divide $4$ by $- 1$ .

$x \geq - 4$

Step 5.6.1.2.8.2

Find the intersection of $x \geq - 4$ and $x < 0$ .

$- 4 \leq x < 0$

Step 5.6.1.2.9

Find the union of the solutions.

$- 4 \leq x \leq 4$

Step 5.6.1.3

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

$[- 4, 4]$

Step 5.6.2

Find the intersection of $y < 0$ and $[- 4, 4]$ .

$- 4 \leq y < 0$

Step 5.7

Write as a piecewise.

${\begin{cases} y > \frac{3 \sqrt{(4 + x) (4 - x)}}{4} & 0 \leq y \leq 4 \\ - y > \frac{3 \sqrt{(4 + x) (4 - x)}}{4} & - 4 \leq y < 0 \end{cases}$

Step 6

Find the intersection of $y > \frac{3 \sqrt{(4 + x) (4 - x)}}{4}$ and $0 \leq y \leq 4$ .

No solution

Step 7

Solve $- y > \frac{3 \sqrt{(4 + x) (4 - x)}}{4}$ when $- 4 \leq y < 0$ .

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Step 7.1

Divide each term in $- y > \frac{3 \sqrt{(4 + x) (4 - x)}}{4}$ by $- 1$ and simplify.

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Step 7.1.1

Divide each term in $- y > \frac{3 \sqrt{(4 + x) (4 - x)}}{4}$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- y}{- 1} < \frac{\frac{3 \sqrt{(4 + x) (4 - x)}}{4}}{- 1}$

Step 7.1.2

Simplify the left side.

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Step 7.1.2.1

Dividing two negative values results in a positive value.

$\frac{y}{1} < \frac{\frac{3 \sqrt{(4 + x) (4 - x)}}{4}}{- 1}$

Step 7.1.2.2

Divide $y$ by $1$ .

$y < \frac{\frac{3 \sqrt{(4 + x) (4 - x)}}{4}}{- 1}$

Step 7.1.3

Simplify the right side.

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Step 7.1.3.1

Move the negative one from the denominator of $\frac{\frac{3 \sqrt{(4 + x) (4 - x)}}{4}}{- 1}$ .

$y < - 1 \cdot \frac{3 \sqrt{(4 + x) (4 - x)}}{4}$

Step 7.1.3.2

Rewrite $- 1 \cdot \frac{3 \sqrt{(4 + x) (4 - x)}}{4}$ as $- \frac{3 \sqrt{(4 + x) (4 - x)}}{4}$ .

$y < - \frac{3 \sqrt{(4 + x) (4 - x)}}{4}$

Step 7.2

Find the intersection of $y < - \frac{3 \sqrt{(4 + x) (4 - x)}}{4}$ and $- 4 \leq y < 0$ .

$- 4 \leq y < N o (Min i m u m)$

Step 8

Find the union of the solutions.

$- 4 \leq y < i N o Min m^{2} u$

Step 9