Pre-Algebra Examples

$\frac{| x - 2 |}{x - 2}$

Step 1

Find the domain for $y = \frac{| x - 2 |}{x - 2}$ 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 $\frac{| x - 2 |}{x - 2}$ equal to $0$ to find where the expression is undefined.

$x - 2 = 0$

Step 1.2

Add $2$ to both sides of the equation.

$x = 2$

Step 1.3

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

Interval Notation:

$(- \infty, 2) \cup (2, \infty)$

Set-Builder Notation:

${x | x \neq 2}$

Interval Notation:

$(- \infty, 2) \cup (2, \infty)$

Set-Builder Notation:

${x | x \neq 2}$

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

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

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

Subtract $2$ from $- 2$ .

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

Step 2.1.2.1.2

The absolute value is the distance between a number and zero. The distance between $- 4$ and $0$ is $4$ .

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

Step 2.1.2.2

Simplify the expression.

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Step 2.1.2.2.1

Subtract $2$ from $- 2$ .

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

Step 2.1.2.2.2

Divide $4$ by $- 4$ .

$f (- 2) = - 1$

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