Algebra Examples Absolute Value Expressions and Equations

Find the Vertex

$y = | \frac{1}{3} x + 1 |$

Step 1

To find the $x$ coordinate of the vertex, set the inside of the absolute value $\frac{1}{3} x + 1$ equal to $0$ . In this case, $\frac{1}{3} x + 1 = 0$ .

$\frac{1}{3} x + 1 = 0$

Step 2

Solve the equation $\frac{1}{3} x + 1 = 0$ to find the $x$ coordinate for the absolute value vertex.

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Step 2.1

Combine $\frac{1}{3}$ and $x$ .

$\frac{x}{3} + 1 = 0$

Step 2.2

Subtract $1$ from both sides of the equation.

$\frac{x}{3} = - 1$

Step 2.3

Multiply both sides of the equation by $3$ .

$3 \frac{x}{3} = 3 \cdot - 1$

Step 2.4

Simplify both sides of the equation.

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Step 2.4.1

Simplify the left side.

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Step 2.4.1.1

Cancel the common factor of $3$ .

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Step 2.4.1.1.1

Cancel the common factor.

$3 \frac{x}{3} = 3 \cdot - 1$

Step 2.4.1.1.2

Rewrite the expression.

$x = 3 \cdot - 1$

Step 2.4.2

Simplify the right side.

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Step 2.4.2.1

Multiply $3$ by $- 1$ .

$x = - 3$

Step 3

Replace the variable $x$ with $- 3$ in the expression.

$y = | \frac{1}{3} \cdot (- 3) + 1 |$

Step 4

Simplify $| \frac{1}{3} \cdot (- 3) + 1 |$ .

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Step 4.1

Cancel the common factor of $3$ .

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Step 4.1.1

Factor $3$ out of $- 3$ .

$y = | \frac{1}{3} \cdot (3 (- 1)) + 1 |$

Step 4.1.2

Cancel the common factor.

$y = | \frac{1}{3} \cdot (3 \cdot - 1) + 1 |$

Step 4.1.3

Rewrite the expression.

$y = | - 1 + 1 |$

Step 4.2

Add $- 1$ and $1$ .

$y = | 0 |$

Step 4.3

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

$y = 0$

Step 5

The absolute value vertex is $(- 3, 0)$ .

$(- 3, 0)$

Step 6

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Find the Vertex

Absolute Value Expressions and Equations

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