Precalculus Examples

$\frac{{(x + 6)}^{2}}{20} + \frac{{(y - 4)}^{2}}{16} = 1$

Step 1

Subtract $\frac{{(x + 6)}^{2}}{20}$ from both sides of the equation.

$\frac{{(y - 4)}^{2}}{16} = 1 - \frac{{(x + 6)}^{2}}{20}$

Step 2

Simplify $1 - \frac{{(x + 6)}^{2}}{20}$ .

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Step 2.1

Combine into one fraction.

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Step 2.1.1

Write $1$ as a fraction with a common denominator.

$\frac{{(y - 4)}^{2}}{16} = \frac{20}{20} - \frac{{(x + 6)}^{2}}{20}$

Step 2.1.2

Combine the numerators over the common denominator.

$\frac{{(y - 4)}^{2}}{16} = \frac{20 - {(x + 6)}^{2}}{20}$

Step 2.2

Simplify the numerator.

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Step 2.2.1

Rewrite ${(x + 6)}^{2}$ as $(x + 6) (x + 6)$ .

$\frac{{(y - 4)}^{2}}{16} = \frac{20 - ((x + 6) (x + 6))}{20}$

Step 2.2.2

Expand $(x + 6) (x + 6)$ using the FOIL Method.

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Step 2.2.2.1

Apply the distributive property.

$\frac{{(y - 4)}^{2}}{16} = \frac{20 - (x (x + 6) + 6 (x + 6))}{20}$

Step 2.2.2.2

Apply the distributive property.

$\frac{{(y - 4)}^{2}}{16} = \frac{20 - (x \cdot x + x \cdot 6 + 6 (x + 6))}{20}$

Step 2.2.2.3

Apply the distributive property.

$\frac{{(y - 4)}^{2}}{16} = \frac{20 - (x \cdot x + x \cdot 6 + 6 x + 6 \cdot 6)}{20}$

Step 2.2.3

Simplify and combine like terms.

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Step 2.2.3.1

Simplify each term.

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Step 2.2.3.1.1

Multiply $x$ by $x$ .

$\frac{{(y - 4)}^{2}}{16} = \frac{20 - (x^{2} + x \cdot 6 + 6 x + 6 \cdot 6)}{20}$

Step 2.2.3.1.2

Move $6$ to the left of $x$ .

$\frac{{(y - 4)}^{2}}{16} = \frac{20 - (x^{2} + 6 \cdot x + 6 x + 6 \cdot 6)}{20}$

Step 2.2.3.1.3

Multiply $6$ by $6$ .

$\frac{{(y - 4)}^{2}}{16} = \frac{20 - (x^{2} + 6 x + 6 x + 36)}{20}$

Step 2.2.3.2

Add $6 x$ and $6 x$ .

$\frac{{(y - 4)}^{2}}{16} = \frac{20 - (x^{2} + 12 x + 36)}{20}$

Step 2.2.4

Apply the distributive property.

$\frac{{(y - 4)}^{2}}{16} = \frac{20 - x^{2} - (12 x) - 1 \cdot 36}{20}$

Step 2.2.5

Simplify.

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Step 2.2.5.1

Multiply $12$ by $- 1$ .

$\frac{{(y - 4)}^{2}}{16} = \frac{20 - x^{2} - 12 x - 1 \cdot 36}{20}$

Step 2.2.5.2

Multiply $- 1$ by $36$ .

$\frac{{(y - 4)}^{2}}{16} = \frac{20 - x^{2} - 12 x - 36}{20}$

Step 2.2.6

Subtract $36$ from $20$ .

$\frac{{(y - 4)}^{2}}{16} = \frac{- x^{2} - 12 x - 16}{20}$

Step 2.3

Simplify with factoring out.

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Step 2.3.1

Factor $- 1$ out of $- x^{2}$ .

$\frac{{(y - 4)}^{2}}{16} = \frac{- (x^{2}) - 12 x - 16}{20}$

Step 2.3.2

Factor $- 1$ out of $- 12 x$ .

$\frac{{(y - 4)}^{2}}{16} = \frac{- (x^{2}) - (12 x) - 16}{20}$

Step 2.3.3

Factor $- 1$ out of $- (x^{2}) - (12 x)$ .

$\frac{{(y - 4)}^{2}}{16} = \frac{- (x^{2} + 12 x) - 16}{20}$

Step 2.3.4

Rewrite $- 16$ as $- 1 (16)$ .

$\frac{{(y - 4)}^{2}}{16} = \frac{- (x^{2} + 12 x) - 1 (16)}{20}$

Step 2.3.5

Factor $- 1$ out of $- (x^{2} + 12 x) - 1 (16)$ .

$\frac{{(y - 4)}^{2}}{16} = \frac{- (x^{2} + 12 x + 16)}{20}$

Step 2.3.6

Simplify the expression.

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Step 2.3.6.1

Rewrite $- (x^{2} + 12 x + 16)$ as $- 1 (x^{2} + 12 x + 16)$ .

$\frac{{(y - 4)}^{2}}{16} = \frac{- 1 (x^{2} + 12 x + 16)}{20}$

Step 2.3.6.2

Move the negative in front of the fraction.

$\frac{{(y - 4)}^{2}}{16} = - \frac{x^{2} + 12 x + 16}{20}$

Step 3

Multiply both sides of the equation by $16$ .

$16 \frac{{(y - 4)}^{2}}{16} = 16 (- \frac{x^{2} + 12 x + 16}{20})$

Step 4

Simplify both sides of the equation.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Cancel the common factor of $16$ .

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Step 4.1.1.1

Cancel the common factor.

$16 \frac{{(y - 4)}^{2}}{16} = 16 (- \frac{x^{2} + 12 x + 16}{20})$

Step 4.1.1.2

Rewrite the expression.

${(y - 4)}^{2} = 16 (- \frac{x^{2} + 12 x + 16}{20})$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $16 (- \frac{x^{2} + 12 x + 16}{20})$ .

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Step 4.2.1.1

Cancel the common factor of $4$ .

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Step 4.2.1.1.1

Move the leading negative in $- \frac{x^{2} + 12 x + 16}{20}$ into the numerator.

${(y - 4)}^{2} = 16 \frac{- (x^{2} + 12 x + 16)}{20}$

Step 4.2.1.1.2

Factor $4$ out of $16$ .

${(y - 4)}^{2} = 4 (4) \frac{- (x^{2} + 12 x + 16)}{20}$

Step 4.2.1.1.3

Factor $4$ out of $20$ .

${(y - 4)}^{2} = 4 \cdot 4 \frac{- (x^{2} + 12 x + 16)}{4 \cdot 5}$

Step 4.2.1.1.4

Cancel the common factor.

${(y - 4)}^{2} = 4 \cdot 4 \frac{- (x^{2} + 12 x + 16)}{4 \cdot 5}$

Step 4.2.1.1.5

Rewrite the expression.

${(y - 4)}^{2} = 4 \frac{- (x^{2} + 12 x + 16)}{5}$

Step 4.2.1.2

Combine $4$ and $\frac{- (x^{2} + 12 x + 16)}{5}$ .

${(y - 4)}^{2} = \frac{4 (- (x^{2} + 12 x + 16))}{5}$

Step 4.2.1.3

Simplify the expression.

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Step 4.2.1.3.1

Multiply $- 1$ by $4$ .

${(y - 4)}^{2} = \frac{- 4 (x^{2} + 12 x + 16)}{5}$

Step 4.2.1.3.2

Move the negative in front of the fraction.

${(y - 4)}^{2} = - \frac{4 (x^{2} + 12 x + 16)}{5}$

Step 5

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

$y - 4 = \pm \sqrt{- \frac{4 (x^{2} + 12 x + 16)}{5}}$

Step 6

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

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Step 6.1

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

$y - 4 = \sqrt{- \frac{4 (x^{2} + 12 x + 16)}{5}}$

Step 6.2

Add $4$ to both sides of the equation.

$y = \sqrt{- \frac{4 (x^{2} + 12 x + 16)}{5}} + 4$

Step 6.3

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

$y - 4 = - \sqrt{- \frac{4 (x^{2} + 12 x + 16)}{5}}$

Step 6.4

Add $4$ to both sides of the equation.

$y = - \sqrt{- \frac{4 (x^{2} + 12 x + 16)}{5}} + 4$

Step 6.5

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

$y = \sqrt{- \frac{4 (x^{2} + 12 x + 16)}{5}} + 4$

$y = - \sqrt{- \frac{4 (x^{2} + 12 x + 16)}{5}} + 4$

$y = \sqrt{- \frac{4 (x^{2} + 12 x + 16)}{5}} + 4$

$y = - \sqrt{- \frac{4 (x^{2} + 12 x + 16)}{5}} + 4$

Step 7

Set the radicand in $\sqrt{- \frac{4 (x^{2} + 12 x + 16)}{5}}$ greater than or equal to $0$ to find where the expression is defined.

$- \frac{4 (x^{2} + 12 x + 16)}{5} \geq 0$

Step 8

Solve for $x$ .

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Step 8.1

Divide each term in $- \frac{4 (x^{2} + 12 x + 16)}{5} \geq 0$ by $- 1$ and simplify.

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Step 8.1.1

Divide each term in $- \frac{4 (x^{2} + 12 x + 16)}{5} \geq 0$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{4 (x^{2} + 12 x + 16)}{5}}{- 1} \leq \frac{0}{- 1}$

Step 8.1.2

Simplify the left side.

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Step 8.1.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{4 (x^{2} + 12 x + 16)}{5}}{1} \leq \frac{0}{- 1}$

Step 8.1.2.2

Divide $\frac{4 (x^{2} + 12 x + 16)}{5}$ by $1$ .

$\frac{4 (x^{2} + 12 x + 16)}{5} \leq \frac{0}{- 1}$

Step 8.1.3

Simplify the right side.

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Step 8.1.3.1

Divide $0$ by $- 1$ .

$\frac{4 (x^{2} + 12 x + 16)}{5} \leq 0$

Step 8.2

Multiply both sides by $5$ .

$\frac{4 (x^{2} + 12 x + 16)}{5} \cdot 5 \leq 0 \cdot 5$

Step 8.3

Simplify.

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Step 8.3.1

Simplify the left side.

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Step 8.3.1.1

Simplify $\frac{4 (x^{2} + 12 x + 16)}{5} \cdot 5$ .

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Step 8.3.1.1.1

Cancel the common factor of $5$ .

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Step 8.3.1.1.1.1

Cancel the common factor.

$\frac{4 (x^{2} + 12 x + 16)}{5} \cdot 5 \leq 0 \cdot 5$

Step 8.3.1.1.1.2

Rewrite the expression.

$4 (x^{2} + 12 x + 16) \leq 0 \cdot 5$

Step 8.3.1.1.2

Apply the distributive property.

$4 x^{2} + 4 (12 x) + 4 \cdot 16 \leq 0 \cdot 5$

Step 8.3.1.1.3

Simplify.

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Step 8.3.1.1.3.1

Multiply $12$ by $4$ .

$4 x^{2} + 48 x + 4 \cdot 16 \leq 0 \cdot 5$

Step 8.3.1.1.3.2

Multiply $4$ by $16$ .

$4 x^{2} + 48 x + 64 \leq 0 \cdot 5$

Step 8.3.2

Simplify the right side.

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Step 8.3.2.1

Multiply $0$ by $5$ .

$4 x^{2} + 48 x + 64 \leq 0$

Step 8.4

Solve for $x$ .

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Step 8.4.1

Convert the inequality to an equation.

$4 x^{2} + 48 x + 64 = 0$

Step 8.4.2

Factor $4$ out of $4 x^{2} + 48 x + 64$ .

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Step 8.4.2.1

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

$4 (x^{2}) + 48 x + 64 = 0$

Step 8.4.2.2

Factor $4$ out of $48 x$ .

$4 (x^{2}) + 4 (12 x) + 64 = 0$

Step 8.4.2.3

Factor $4$ out of $64$ .

$4 x^{2} + 4 (12 x) + 4 \cdot 16 = 0$

Step 8.4.2.4

Factor $4$ out of $4 x^{2} + 4 (12 x)$ .

$4 (x^{2} + 12 x) + 4 \cdot 16 = 0$

Step 8.4.2.5

Factor $4$ out of $4 (x^{2} + 12 x) + 4 \cdot 16$ .

$4 (x^{2} + 12 x + 16) = 0$

Step 8.4.3

Divide each term in $4 (x^{2} + 12 x + 16) = 0$ by $4$ and simplify.

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Step 8.4.3.1

Divide each term in $4 (x^{2} + 12 x + 16) = 0$ by $4$ .

$\frac{4 (x^{2} + 12 x + 16)}{4} = \frac{0}{4}$

Step 8.4.3.2

Simplify the left side.

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Step 8.4.3.2.1

Cancel the common factor of $4$ .

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Step 8.4.3.2.1.1

Cancel the common factor.

$\frac{4 (x^{2} + 12 x + 16)}{4} = \frac{0}{4}$

Step 8.4.3.2.1.2

Divide $x^{2} + 12 x + 16$ by $1$ .

$x^{2} + 12 x + 16 = \frac{0}{4}$

Step 8.4.3.3

Simplify the right side.

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Step 8.4.3.3.1

Divide $0$ by $4$ .

$x^{2} + 12 x + 16 = 0$

Step 8.4.4

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 8.4.5

Substitute the values $a = 1$ , $b = 12$ , and $c = 16$ into the quadratic formula and solve for $x$ .

$\frac{- 12 \pm \sqrt{12^{2} - 4 \cdot (1 \cdot 16)}}{2 \cdot 1}$

Step 8.4.6

Simplify.

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Step 8.4.6.1

Simplify the numerator.

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Step 8.4.6.1.1

Raise $12$ to the power of $2$ .

$x = \frac{- 12 \pm \sqrt{144 - 4 \cdot 1 \cdot 16}}{2 \cdot 1}$

Step 8.4.6.1.2

Multiply $- 4 \cdot 1 \cdot 16$ .

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Step 8.4.6.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 12 \pm \sqrt{144 - 4 \cdot 16}}{2 \cdot 1}$

Step 8.4.6.1.2.2

Multiply $- 4$ by $16$ .

$x = \frac{- 12 \pm \sqrt{144 - 64}}{2 \cdot 1}$

Step 8.4.6.1.3

Subtract $64$ from $144$ .

$x = \frac{- 12 \pm \sqrt{80}}{2 \cdot 1}$

Step 8.4.6.1.4

Rewrite $80$ as $4^{2} \cdot 5$ .

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Step 8.4.6.1.4.1

Factor $16$ out of $80$ .

$x = \frac{- 12 \pm \sqrt{16 (5)}}{2 \cdot 1}$

Step 8.4.6.1.4.2

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

$x = \frac{- 12 \pm \sqrt{4^{2} \cdot 5}}{2 \cdot 1}$

Step 8.4.6.1.5

Pull terms out from under the radical.

$x = \frac{- 12 \pm 4 \sqrt{5}}{2 \cdot 1}$

Step 8.4.6.2

Multiply $2$ by $1$ .

$x = \frac{- 12 \pm 4 \sqrt{5}}{2}$

Step 8.4.6.3

Simplify $\frac{- 12 \pm 4 \sqrt{5}}{2}$ .

$x = - 6 \pm 2 \sqrt{5}$

Step 8.4.7

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 8.4.7.1

Simplify the numerator.

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Step 8.4.7.1.1

Raise $12$ to the power of $2$ .

$x = \frac{- 12 \pm \sqrt{144 - 4 \cdot 1 \cdot 16}}{2 \cdot 1}$

Step 8.4.7.1.2

Multiply $- 4 \cdot 1 \cdot 16$ .

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Step 8.4.7.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 12 \pm \sqrt{144 - 4 \cdot 16}}{2 \cdot 1}$

Step 8.4.7.1.2.2

Multiply $- 4$ by $16$ .

$x = \frac{- 12 \pm \sqrt{144 - 64}}{2 \cdot 1}$

Step 8.4.7.1.3

Subtract $64$ from $144$ .

$x = \frac{- 12 \pm \sqrt{80}}{2 \cdot 1}$

Step 8.4.7.1.4

Rewrite $80$ as $4^{2} \cdot 5$ .

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Step 8.4.7.1.4.1

Factor $16$ out of $80$ .

$x = \frac{- 12 \pm \sqrt{16 (5)}}{2 \cdot 1}$

Step 8.4.7.1.4.2

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

$x = \frac{- 12 \pm \sqrt{4^{2} \cdot 5}}{2 \cdot 1}$

Step 8.4.7.1.5

Pull terms out from under the radical.

$x = \frac{- 12 \pm 4 \sqrt{5}}{2 \cdot 1}$

Step 8.4.7.2

Multiply $2$ by $1$ .

$x = \frac{- 12 \pm 4 \sqrt{5}}{2}$

Step 8.4.7.3

Simplify $\frac{- 12 \pm 4 \sqrt{5}}{2}$ .

$x = - 6 \pm 2 \sqrt{5}$

Step 8.4.7.4

Change the $\pm$ to $+$ .

$x = - 6 + 2 \sqrt{5}$

Step 8.4.8

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 8.4.8.1

Simplify the numerator.

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Step 8.4.8.1.1

Raise $12$ to the power of $2$ .

$x = \frac{- 12 \pm \sqrt{144 - 4 \cdot 1 \cdot 16}}{2 \cdot 1}$

Step 8.4.8.1.2

Multiply $- 4 \cdot 1 \cdot 16$ .

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Step 8.4.8.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 12 \pm \sqrt{144 - 4 \cdot 16}}{2 \cdot 1}$

Step 8.4.8.1.2.2

Multiply $- 4$ by $16$ .

$x = \frac{- 12 \pm \sqrt{144 - 64}}{2 \cdot 1}$

Step 8.4.8.1.3

Subtract $64$ from $144$ .

$x = \frac{- 12 \pm \sqrt{80}}{2 \cdot 1}$

Step 8.4.8.1.4

Rewrite $80$ as $4^{2} \cdot 5$ .

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Step 8.4.8.1.4.1

Factor $16$ out of $80$ .

$x = \frac{- 12 \pm \sqrt{16 (5)}}{2 \cdot 1}$

Step 8.4.8.1.4.2

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

$x = \frac{- 12 \pm \sqrt{4^{2} \cdot 5}}{2 \cdot 1}$

Step 8.4.8.1.5

Pull terms out from under the radical.

$x = \frac{- 12 \pm 4 \sqrt{5}}{2 \cdot 1}$

Step 8.4.8.2

Multiply $2$ by $1$ .

$x = \frac{- 12 \pm 4 \sqrt{5}}{2}$

Step 8.4.8.3

Simplify $\frac{- 12 \pm 4 \sqrt{5}}{2}$ .

$x = - 6 \pm 2 \sqrt{5}$

Step 8.4.8.4

Change the $\pm$ to $-$ .

$x = - 6 - 2 \sqrt{5}$

Step 8.4.9

The final answer is the combination of both solutions.

$x = - 6 + 2 \sqrt{5}, - 6 - 2 \sqrt{5}$

Step 8.5

Use each root to create test intervals.

$x < - 6 - 2 \sqrt{5}$

$- 6 - 2 \sqrt{5} < x < - 6 + 2 \sqrt{5}$

$x > - 6 + 2 \sqrt{5}$

Step 8.6

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

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Step 8.6.1

Test a value on the interval $x < - 6 - 2 \sqrt{5}$ to see if it makes the inequality true.

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Step 8.6.1.1

Choose a value on the interval $x < - 6 - 2 \sqrt{5}$ and see if this value makes the original inequality true.

$x = - 13$

Step 8.6.1.2

Replace $x$ with $- 13$ in the original inequality.

$- \frac{4 ({(- 13)}^{2} + 12 (- 13) + 16)}{5} \geq 0$

Step 8.6.1.3

The left side $- 23.2$ is less than the right side $0$ , which means that the given statement is false.

False

Step 8.6.2

Test a value on the interval $- 6 - 2 \sqrt{5} < x < - 6 + 2 \sqrt{5}$ to see if it makes the inequality true.

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Step 8.6.2.1

Choose a value on the interval $- 6 - 2 \sqrt{5} < x < - 6 + 2 \sqrt{5}$ and see if this value makes the original inequality true.

$x = - 6$

Step 8.6.2.2

Replace $x$ with $- 6$ in the original inequality.

$- \frac{4 ({(- 6)}^{2} + 12 (- 6) + 16)}{5} \geq 0$

Step 8.6.2.3

The left side $16$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 8.6.3

Test a value on the interval $x > - 6 + 2 \sqrt{5}$ to see if it makes the inequality true.

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Step 8.6.3.1

Choose a value on the interval $x > - 6 + 2 \sqrt{5}$ and see if this value makes the original inequality true.

$x = 0$

Step 8.6.3.2

Replace $x$ with $0$ in the original inequality.

$- \frac{4 ({(0)}^{2} + 12 (0) + 16)}{5} \geq 0$

Step 8.6.3.3

The left side $- 12.8$ is less than the right side $0$ , which means that the given statement is false.

False

Step 8.6.4

Compare the intervals to determine which ones satisfy the original inequality.

$x < - 6 - 2 \sqrt{5}$ False

$- 6 - 2 \sqrt{5} < x < - 6 + 2 \sqrt{5}$ True

$x > - 6 + 2 \sqrt{5}$ False

$x < - 6 - 2 \sqrt{5}$ False

$- 6 - 2 \sqrt{5} < x < - 6 + 2 \sqrt{5}$ True

$x > - 6 + 2 \sqrt{5}$ False

Step 8.7

The solution consists of all of the true intervals.

$- 6 - 2 \sqrt{5} \leq x \leq - 6 + 2 \sqrt{5}$

Step 9

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

Interval Notation:

$[- 6 - 2 \sqrt{5}, - 6 + 2 \sqrt{5}]$

Set-Builder Notation:

${x | - 6 - 2 \sqrt{5} \leq x \leq - 6 + 2 \sqrt{5}}$

Step 10

The range is the set of all valid $y$ values. Use the graph to find the range.

Interval Notation:

$[0, 8]$

Set-Builder Notation:

${y | 0 \leq y \leq 8}$

Step 11

Determine the domain and range.

Domain: $[- 6 - 2 \sqrt{5}, - 6 + 2 \sqrt{5}], {x | - 6 - 2 \sqrt{5} \leq x \leq - 6 + 2 \sqrt{5}}$

Range: $[0, 8], {y | 0 \leq y \leq 8}$

Step 12