Algebra Examples

$3 x + | y | = 0$

Step 1

Subtract $| y |$ from both sides of the equation.

$3 x = - | y |$

Step 2

Divide each term in $3 x = - | y |$ by $3$ and simplify.

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Step 2.1

Divide each term in $3 x = - | y |$ by $3$ .

$\frac{3 x}{3} = \frac{- | y |}{3}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $3$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{- | y |}{3}$

Step 2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- | y |}{3}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Move the negative in front of the fraction.

$x = - \frac{| y |}{3}$

Step 3

Find the absolute value vertex. In this case, the vertex for $x = - \frac{| y |}{3}$ is $(0, 0)$ .

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Step 3.1

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 3.2

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

$x = - \frac{| 0 |}{3}$

Step 3.3

Simplify $- \frac{| 0 |}{3}$ .

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Step 3.3.1

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

$x = - \frac{0}{3}$

Step 3.3.2

Divide $0$ by $3$ .

$x = - 0$

Step 3.3.3

Multiply $- 1$ by $0$ .

$x = 0$

Step 3.4

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

$(0, 0)$

Step 4

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

$(- \infty, 0]$

${x | x \leq 0}$

Step 5

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 5.1

Substitute the $x$ value $- 2$ into $f (x) = - 3 x$ . In this case, the point is $(- 2, 6)$ .

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Step 5.1.1

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

$f (- 2) = - 3 \cdot - 2$

Step 5.1.2

Simplify the result.

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Step 5.1.2.1

Multiply $- 3$ by $- 2$ .

$f (- 2) = 6$

Step 5.1.2.2

The final answer is $6$ .

$y = 6$

Step 5.2

Substitute the $x$ value $- 2$ into $f (x) = 3 x$ . In this case, the point is $(- 2, - 6)$ .

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Step 5.2.1

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

$f (- 2) = 3 (- 2)$

Step 5.2.2

Simplify the result.

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Step 5.2.2.1

Multiply $3$ by $- 2$ .

$f (- 2) = - 6$

Step 5.2.2.2

The final answer is $- 6$ .

$y = - 6$

Step 5.3

Substitute the $x$ value $- 1$ into $f (x) = - 3 x$ . In this case, the point is $(- 1, 3)$ .

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Step 5.3.1

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

$f (- 1) = - 3 \cdot - 1$

Step 5.3.2

Simplify the result.

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Step 5.3.2.1

Multiply $- 3$ by $- 1$ .

$f (- 1) = 3$

Step 5.3.2.2

The final answer is $3$ .

$y = 3$

Step 5.4

Substitute the $x$ value $- 1$ into $f (x) = 3 x$ . In this case, the point is $(- 1, - 3)$ .

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Step 5.4.1

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

$f (- 1) = 3 (- 1)$

Step 5.4.2

Simplify the result.

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Step 5.4.2.1

Multiply $3$ by $- 1$ .

$f (- 1) = - 3$

Step 5.4.2.2

The final answer is $- 3$ .

$y = - 3$

Step 5.5

The absolute value can be graphed using the points around the vertex $(0, 0), (- 2, 6), (- 2, - 6), (- 1, 3), (- 1, - 3)$

$\begin{array}{c} x & y \\ - 2 & 6 \\ - 2 & - 6 \\ - 1 & 3 \\ - 1 & - 3 \\ 0 & 0 \end{array}$

Step 6