Pre-Algebra Examples

- \frac{| x |}{x}

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

Step 1

Find the domain for

y = - \frac{| x |}{x}

$y = - \frac{| x |}{x}$ so that a list of

x

$x$ 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}

$\frac{| x |}{x}$ equal to

0

$0$ to find where the expression is undefined.

x = 0

$x = 0$

Step 1.2

The domain is all values of

x

$x$ that make the expression defined.

Interval Notation:

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

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

Set-Builder Notation:

{x | x \neq 0}

${x | x \neq 0}$

Interval Notation:

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

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

Set-Builder Notation:

{x | x \neq 0}

${x | x \neq 0}$

Step 2

For each

x

$x$ value, there is one

y

$y$ value. Select a few

x

$x$ values from the domain. It would be more useful to select the values so that they are around the

x

$x$ value of the absolute value vertex.

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Step 2.1

Substitute the

x

$x$ value

- 2

$- 2$ into

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

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

(- 2, 1)

$(- 2, 1)$ .

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Step 2.1.1

Replace the variable

x

$x$ with

- 2

$- 2$ in the expression.

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

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

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

$- 2$ and

0

$0$ is

2

$2$ .

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

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

Step 2.1.2.2

Divide

2

$2$ by

- 2

$- 2$ .

f (- 2) = 1

$f (- 2) = 1$

Step 2.1.2.3

The final answer is

1

$1$ .

y = 1

$y = 1$

y = 1

$y = 1$

y = 1

$y = 1$

Step 2.2

Substitute the

x

$x$ value

1

$1$ into

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

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

(1, - 1)

$(1, - 1)$ .

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Step 2.2.1

Replace the variable

x

$x$ with

1

$1$ in the expression.

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

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

Step 2.2.2

Simplify the result.

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Step 2.2.2.1

Divide

| 1 |

$| 1 |$ by

1

$1$ .

f (1) = - | 1 |

$f (1) = - | 1 |$

Step 2.2.2.2

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

0

$0$ and

1

$1$ is

1

$1$ .

f (1) = - 1 \cdot 1

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

Step 2.2.2.3

Multiply

- 1

$- 1$ by

1

$1$ .

f (1) = - 1

$f (1) = - 1$

Step 2.2.2.4

The final answer is

- 1

$- 1$ .

y = - 1

$y = - 1$

y = - 1

$y = - 1$

y = - 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)

$(- 2, 1), (- 1, 1), (1, - 1), (2, - 1)$

\begin{matrix} x & y - 2 & 1 - 1 & 1 1 & - 1 2 & - 1 \end{matrix}

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

\begin{matrix} x & y - 2 & 1 - 1 & 1 1 & - 1 2 & - 1 \end{matrix}

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

Step 3