Pre-Algebra Examples

$- \frac{| x |}{x}$

Step 1

Find the domain for $y = - \frac{| x |}{x}$ 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 $\frac{| x |}{x}$ equal to $0$ to find where the expression is undefined.

$x = 0$

Step 1.2

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

Interval Notation:

$(- \infty, 0) \cup (0, \infty)$

Set-Builder Notation:

${x | x \neq 0}$

Interval Notation:

$(- \infty, 0) \cup (0, \infty)$

Set-Builder Notation:

${x | x \neq 0}$

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 $f (x) = - \frac{| x |}{x}$ . 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.

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

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

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

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

Step 2.1.2.2

Divide $2$ by $- 2$ .

$f (- 2) = 1$

Step 2.1.2.3

The final answer is $1$ .

$y = 1$

Step 2.2

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

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Step 2.2.1

Replace the variable $x$ with $1$ in the expression.

$f (1) = - \frac{| 1 |}{1}$

Step 2.2.2

Simplify the result.

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Step 2.2.2.1

Divide $| 1 |$ by $1$ .

$f (1) = - | 1 |$

Step 2.2.2.2

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$f (1) = - 1 \cdot 1$

Step 2.2.2.3

Multiply $- 1$ by $1$ .

$f (1) = - 1$

Step 2.2.2.4

The final answer is $- 1$ .

$y = - 1$

Step 2.3

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

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

Step 3