Pre-Algebra Examples

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

Step 1

Find the $y$ value at $x = - 12$ .

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Step 1.1

Replace the variable $x$ with $- 12$ in the expression.

$f (- 12) = \frac{\sqrt{(3 (- 12) - 4) ((- 12) + 2)}}{(- 12) + 2}$

Step 1.2

Simplify the result.

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Step 1.2.1

Simplify the numerator.

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Step 1.2.1.1

Multiply $3$ by $- 12$ .

$f (- 12) = \frac{\sqrt{(- 36 - 4) (- 12 + 2)}}{- 12 + 2}$

Step 1.2.1.2

Subtract $4$ from $- 36$ .

$f (- 12) = \frac{\sqrt{- 40 (- 12 + 2)}}{- 12 + 2}$

Step 1.2.1.3

Add $- 12$ and $2$ .

$f (- 12) = \frac{\sqrt{- 40 \cdot - 10}}{- 12 + 2}$

Step 1.2.1.4

Multiply $- 40$ by $- 10$ .

$f (- 12) = \frac{\sqrt{400}}{- 12 + 2}$

Step 1.2.1.5

Rewrite $400$ as $20^{2}$ .

$f (- 12) = \frac{\sqrt{20^{2}}}{- 12 + 2}$

Step 1.2.1.6

Pull terms out from under the radical, assuming positive real numbers.

$f (- 12) = \frac{20}{- 12 + 2}$

Step 1.2.2

Simplify the expression.

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Step 1.2.2.1

Add $- 12$ and $2$ .

$f (- 12) = \frac{20}{- 10}$

Step 1.2.2.2

Divide $20$ by $- 10$ .

$f (- 12) = - 2$

Step 1.2.3

The final answer is $- 2$ .

$- 2$

Step 1.3

The $y$ value at $x = - 12$ is $- 2$ .

$y = - 2$

Step 2

Find the $y$ value at $x = - 3$ .

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Step 2.1

Replace the variable $x$ with $- 3$ in the expression.

$f (- 3) = \frac{\sqrt{(3 (- 3) - 4) ((- 3) + 2)}}{(- 3) + 2}$

Step 2.2

Simplify the result.

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Step 2.2.1

Simplify the numerator.

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Step 2.2.1.1

Multiply $3$ by $- 3$ .

$f (- 3) = \frac{\sqrt{(- 9 - 4) (- 3 + 2)}}{- 3 + 2}$

Step 2.2.1.2

Subtract $4$ from $- 9$ .

$f (- 3) = \frac{\sqrt{- 13 (- 3 + 2)}}{- 3 + 2}$

Step 2.2.1.3

Add $- 3$ and $2$ .

$f (- 3) = \frac{\sqrt{- 13 \cdot - 1}}{- 3 + 2}$

Step 2.2.1.4

Multiply $- 13$ by $- 1$ .

$f (- 3) = \frac{\sqrt{13}}{- 3 + 2}$

Step 2.2.2

Simplify the expression.

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Step 2.2.2.1

Add $- 3$ and $2$ .

$f (- 3) = \frac{\sqrt{13}}{- 1}$

Step 2.2.2.2

Move the negative one from the denominator of $\frac{\sqrt{13}}{- 1}$ .

$f (- 3) = - 1 \cdot \sqrt{13}$

Step 2.2.2.3

Rewrite $- 1 \cdot \sqrt{13}$ as $- \sqrt{13}$ .

$f (- 3) = - \sqrt{13}$

Step 2.2.3

The final answer is $- \sqrt{13}$ .

$- \sqrt{13}$

Step 2.3

The $y$ value at $x = - 3$ is $- \sqrt{13}$ .

$y = - \sqrt{13}$

Step 3

Find the $y$ value at $x = - 4$ .

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Step 3.1

Replace the variable $x$ with $- 4$ in the expression.

$f (- 4) = \frac{\sqrt{(3 (- 4) - 4) ((- 4) + 2)}}{(- 4) + 2}$

Step 3.2

Simplify the result.

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Step 3.2.1

Simplify the numerator.

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Step 3.2.1.1

Multiply $3$ by $- 4$ .

$f (- 4) = \frac{\sqrt{(- 12 - 4) (- 4 + 2)}}{- 4 + 2}$

Step 3.2.1.2

Subtract $4$ from $- 12$ .

$f (- 4) = \frac{\sqrt{- 16 (- 4 + 2)}}{- 4 + 2}$

Step 3.2.1.3

Add $- 4$ and $2$ .

$f (- 4) = \frac{\sqrt{- 16 \cdot - 2}}{- 4 + 2}$

Step 3.2.1.4

Multiply $- 16$ by $- 2$ .

$f (- 4) = \frac{\sqrt{32}}{- 4 + 2}$

Step 3.2.1.5

Rewrite $32$ as $4^{2} \cdot 2$ .

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Step 3.2.1.5.1

Factor $16$ out of $32$ .

$f (- 4) = \frac{\sqrt{16 (2)}}{- 4 + 2}$

Step 3.2.1.5.2

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

$f (- 4) = \frac{\sqrt{4^{2} \cdot 2}}{- 4 + 2}$

Step 3.2.1.6

Pull terms out from under the radical.

$f (- 4) = \frac{4 \sqrt{2}}{- 4 + 2}$

Step 3.2.2

Reduce the expression by cancelling the common factors.

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Step 3.2.2.1

Add $- 4$ and $2$ .

$f (- 4) = \frac{4 \sqrt{2}}{- 2}$

Step 3.2.2.2

Cancel the common factor of $4$ and $- 2$ .

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Step 3.2.2.2.1

Factor $2$ out of $4 \sqrt{2}$ .

$f (- 4) = \frac{2 (2 \sqrt{2})}{- 2}$

Step 3.2.2.2.2

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

$f (- 4) = - 1 \cdot (2 \sqrt{2})$

Step 3.2.2.3

Simplify the expression.

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Step 3.2.2.3.1

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

$f (- 4) = - (2 \sqrt{2})$

Step 3.2.2.3.2

Multiply $2$ by $- 1$ .

$f (- 4) = - 2 \sqrt{2}$

Step 3.2.3

The final answer is $- 2 \sqrt{2}$ .

$- 2 \sqrt{2}$

Step 3.3

The $y$ value at $x = - 4$ is $- 2 \sqrt{2}$ .

$y = - 2 \sqrt{2}$

Step 4

Find the $y$ value at $x = 3$ .

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Step 4.1

Replace the variable $x$ with $3$ in the expression.

$f (3) = \frac{\sqrt{(3 (3) - 4) ((3) + 2)}}{(3) + 2}$

Step 4.2

Simplify the result.

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Step 4.2.1

Simplify the numerator.

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Step 4.2.1.1

Multiply $3$ by $3$ .

$f (3) = \frac{\sqrt{(9 - 4) (3 + 2)}}{3 + 2}$

Step 4.2.1.2

Subtract $4$ from $9$ .

$f (3) = \frac{\sqrt{5 (3 + 2)}}{3 + 2}$

Step 4.2.1.3

Add $3$ and $2$ .

$f (3) = \frac{\sqrt{5 \cdot 5}}{3 + 2}$

Step 4.2.1.4

Multiply $5$ by $5$ .

$f (3) = \frac{\sqrt{25}}{3 + 2}$

Step 4.2.1.5

Rewrite $25$ as $5^{2}$ .

$f (3) = \frac{\sqrt{5^{2}}}{3 + 2}$

Step 4.2.1.6

Pull terms out from under the radical, assuming positive real numbers.

$f (3) = \frac{5}{3 + 2}$

Step 4.2.2

Simplify the expression.

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Step 4.2.2.1

Add $3$ and $2$ .

$f (3) = \frac{5}{5}$

Step 4.2.2.2

Divide $5$ by $5$ .

$f (3) = 1$

Step 4.2.3

The final answer is $1$ .

$1$

Step 4.3

The $y$ value at $x = 3$ is $1$ .

$y = 1$

Step 5

Find the $y$ value at $x = 2$ .

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Step 5.1

Replace the variable $x$ with $2$ in the expression.

$f (2) = \frac{\sqrt{(3 (2) - 4) ((2) + 2)}}{(2) + 2}$

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify the numerator.

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Step 5.2.1.1

Multiply $3$ by $2$ .

$f (2) = \frac{\sqrt{(6 - 4) (2 + 2)}}{2 + 2}$

Step 5.2.1.2

Subtract $4$ from $6$ .

$f (2) = \frac{\sqrt{2 (2 + 2)}}{2 + 2}$

Step 5.2.1.3

Add $2$ and $2$ .

$f (2) = \frac{\sqrt{2 \cdot 4}}{2 + 2}$

Step 5.2.1.4

Multiply $2$ by $4$ .

$f (2) = \frac{\sqrt{8}}{2 + 2}$

Step 5.2.1.5

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 5.2.1.5.1

Factor $4$ out of $8$ .

$f (2) = \frac{\sqrt{4 (2)}}{2 + 2}$

Step 5.2.1.5.2

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

$f (2) = \frac{\sqrt{2^{2} \cdot 2}}{2 + 2}$

Step 5.2.1.6

Pull terms out from under the radical.

$f (2) = \frac{2 \sqrt{2}}{2 + 2}$

Step 5.2.2

Reduce the expression by cancelling the common factors.

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Step 5.2.2.1

Add $2$ and $2$ .

$f (2) = \frac{2 \sqrt{2}}{4}$

Step 5.2.2.2

Cancel the common factor of $2$ and $4$ .

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Step 5.2.2.2.1

Factor $2$ out of $2 \sqrt{2}$ .

$f (2) = \frac{2 (\sqrt{2})}{4}$

Step 5.2.2.2.2

Cancel the common factors.

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Step 5.2.2.2.2.1

Factor $2$ out of $4$ .

$f (2) = \frac{2 \sqrt{2}}{2 \cdot 2}$

Step 5.2.2.2.2.2

Cancel the common factor.

$f (2) = \frac{2 \sqrt{2}}{2 \cdot 2}$

Step 5.2.2.2.2.3

Rewrite the expression.

$f (2) = \frac{\sqrt{2}}{2}$

Step 5.2.3

The final answer is $\frac{\sqrt{2}}{2}$ .

$\frac{\sqrt{2}}{2}$

Step 5.3

The $y$ value at $x = 2$ is $\frac{\sqrt{2}}{2}$ .

$y = \frac{\sqrt{2}}{2}$

Step 6

Find the $y$ value at $x = 4$ .

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Step 6.1

Replace the variable $x$ with $4$ in the expression.

$f (4) = \frac{\sqrt{(3 (4) - 4) ((4) + 2)}}{(4) + 2}$

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify the numerator.

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Step 6.2.1.1

Multiply $3$ by $4$ .

$f (4) = \frac{\sqrt{(12 - 4) (4 + 2)}}{4 + 2}$

Step 6.2.1.2

Subtract $4$ from $12$ .

$f (4) = \frac{\sqrt{8 (4 + 2)}}{4 + 2}$

Step 6.2.1.3

Add $4$ and $2$ .

$f (4) = \frac{\sqrt{8 \cdot 6}}{4 + 2}$

Step 6.2.1.4

Multiply $8$ by $6$ .

$f (4) = \frac{\sqrt{48}}{4 + 2}$

Step 6.2.1.5

Rewrite $48$ as $4^{2} \cdot 3$ .

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Step 6.2.1.5.1

Factor $16$ out of $48$ .

$f (4) = \frac{\sqrt{16 (3)}}{4 + 2}$

Step 6.2.1.5.2

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

$f (4) = \frac{\sqrt{4^{2} \cdot 3}}{4 + 2}$

Step 6.2.1.6

Pull terms out from under the radical.

$f (4) = \frac{4 \sqrt{3}}{4 + 2}$

Step 6.2.2

Reduce the expression by cancelling the common factors.

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Step 6.2.2.1

Add $4$ and $2$ .

$f (4) = \frac{4 \sqrt{3}}{6}$

Step 6.2.2.2

Cancel the common factor of $4$ and $6$ .

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Step 6.2.2.2.1

Factor $2$ out of $4 \sqrt{3}$ .

$f (4) = \frac{2 (2 \sqrt{3})}{6}$

Step 6.2.2.2.2

Cancel the common factors.

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Step 6.2.2.2.2.1

Factor $2$ out of $6$ .

$f (4) = \frac{2 (2 \sqrt{3})}{2 (3)}$

Step 6.2.2.2.2.2

Cancel the common factor.

$f (4) = \frac{2 (2 \sqrt{3})}{2 \cdot 3}$

Step 6.2.2.2.2.3

Rewrite the expression.

$f (4) = \frac{2 \sqrt{3}}{3}$

Step 6.2.3

The final answer is $\frac{2 \sqrt{3}}{3}$ .

$\frac{2 \sqrt{3}}{3}$

Step 6.3

The $y$ value at $x = 4$ is $\frac{2 \sqrt{3}}{3}$ .

$y = \frac{2 \sqrt{3}}{3}$

Step 7

List the points to graph.

$(- 12, - 2), (- 3, - 3.60555127), (- 4, - 2.82842712), (3, 1), (2, 0.70710678), (4, 1.15470053)$

Step 8

Select a few points to graph.

$\begin{array}{c} x & y \\ - 12 & - 2 \\ - 4 & - 2.828 \\ - 3 & - 3.606 \\ 2 & 0.707 \\ 3 & 1 \\ 4 & 1.155 \end{array}$

Step 9