Algebra Examples

$| y | = \frac{x}{2}$

Step 1

Rewrite the equation as $\frac{x}{2} = | y |$ .

$\frac{x}{2} = | y |$

Step 2

Multiply both sides of the equation by $2$ .

$2 \frac{x}{2} = 2 | y |$

Step 3

Simplify the left side.

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Step 3.1

Cancel the common factor of $2$ .

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Step 3.1.1

Cancel the common factor.

$2 \frac{x}{2} = 2 | y |$

Step 3.1.2

Rewrite the expression.

$x = 2 | y |$

Step 4

Find the absolute value vertex. In this case, the vertex for $x = 2 | y |$ is $(0, 0)$ .

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Step 4.1

To find the $y$ coordinate of the vertex, set the inside of the absolute value $y$ equal to $0$ . In this case, $y = 0$ .

$y = 0$

Step 4.2

Replace the variable $y$ with $0$ in the expression.

$x = 2 | 0 |$

Step 4.3

Simplify $2 | 0 |$ .

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Step 4.3.1

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

$x = 2 \cdot 0$

Step 4.3.2

Multiply $2$ by $0$ .

$x = 0$

Step 4.4

The absolute value vertex is $(0, 0)$ .

$(0, 0)$

Step 5

The domain is the set of all valid $x$ values. Use the graph to find the domain.

$[0, \infty)$

${x | x \geq 0}$

Step 6

For each $x$ value, there is one positive $y$ value and one negative $y$ 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 6.1

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

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Step 6.1.1

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

$f (1) = \frac{1}{2}$

Step 6.1.2

The final answer is $\frac{1}{2}$ .

$y = \frac{1}{2}$

Step 6.2

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

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Step 6.2.1

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

$f (1) = - \frac{1}{2}$

Step 6.2.2

The final answer is $- \frac{1}{2}$ .

$y = - \frac{1}{2}$

Step 6.3

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

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Step 6.3.1

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

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

Step 6.3.2

Simplify the result.

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Step 6.3.2.1

Divide $2$ by $2$ .

$f (2) = 1$

Step 6.3.2.2

The final answer is $1$ .

$y = 1$

Step 6.4

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

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Step 6.4.1

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

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

Step 6.4.2

Simplify the result.

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Step 6.4.2.1

Cancel the common factor of $2$ .

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Step 6.4.2.1.1

Cancel the common factor.

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

Step 6.4.2.1.2

Rewrite the expression.

$f (2) = - 1 \cdot 1$

Step 6.4.2.2

Multiply $- 1$ by $1$ .

$f (2) = - 1$

Step 6.4.2.3

The final answer is $- 1$ .

$y = - 1$

Step 6.5

The absolute value can be graphed using the points around the vertex $(0, 0), (1, 0.5), (1, - 0.5), (2, 1), (2, - 1)$

$\begin{array}{c} x & y \\ 0 & 0 \\ 1 & 0.5 \\ 1 & - 0.5 \\ 2 & 1 \\ 2 & - 1 \end{array}$

Step 7