Pre-Algebra Examples

$\frac{| x + 8 |}{x + 8}$

Step 1

Find the domain for $y = \frac{| x + 8 |}{x + 8}$ so that a list of $x$ values can be picked to find a list of points, which will help graphing the absolute value function.

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Step 1.1

Set the denominator in $\frac{| x + 8 |}{x + 8}$ equal to $0$ to find where the expression is undefined.

$x + 8 = 0$

Step 1.2

Subtract $8$ from both sides of the equation.

$x = - 8$

Step 1.3

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

Interval Notation:

$(- \infty, - 8) \cup (- 8, \infty)$

Set-Builder Notation:

${x | x \neq - 8}$

Interval Notation:

$(- \infty, - 8) \cup (- 8, \infty)$

Set-Builder Notation:

${x | x \neq - 8}$

Step 2

For each $x$ value, there is one $y$ value. Select a few $x$ values from the domain. It would be more useful to select the values so that they are around the $x$ value of the absolute value vertex.

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Step 2.1

Substitute the $x$ value $- 2$ into $f (x) = \frac{| x + 8 |}{x + 8}$ . In this case, the point is $(- 2, 1)$ .

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Step 2.1.1

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

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

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Simplify the numerator.

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Step 2.1.2.1.1

Add $- 2$ and $8$ .

$f (- 2) = \frac{| 6 |}{- 2 + 8}$

Step 2.1.2.1.2

The absolute value is the distance between a number and zero. The distance between $0$ and $6$ is $6$ .

$f (- 2) = \frac{6}{- 2 + 8}$

Step 2.1.2.2

Simplify the expression.

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Step 2.1.2.2.1

Add $- 2$ and $8$ .

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

Step 2.1.2.2.2

Divide $6$ by $6$ .

$f (- 2) = 1$

Step 2.1.2.3

The final answer is $1$ .

$y = 1$

Step 2.2

The absolute value can be graphed using the points around the vertex $(- 2, 1), (- 1, 1), (0, 1), (1, 1)$

$\begin{array}{c} x & y \\ - 2 & 1 \\ - 1 & 1 \\ 0 & 1 \\ 1 & 1 \end{array}$

Step 3