Algebra Examples

$f (x) = - | \frac{1}{3} (x + 3) |$

Step 1

Find the x-intercepts.

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Step 1.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = - | \frac{1}{3} \cdot (x + 3) |$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $- | \frac{1}{3} \cdot (x + 3) | = 0$ .

$- | \frac{1}{3} \cdot (x + 3) | = 0$

Step 1.2.2

Divide each term in $- | \frac{1}{3} \cdot (x + 3) | = 0$ by $- 1$ and simplify.

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Step 1.2.2.1

Divide each term in $- | \frac{1}{3} \cdot (x + 3) | = 0$ by $- 1$ .

$\frac{- | \frac{1}{3} \cdot (x + 3) |}{- 1} = \frac{0}{- 1}$

Step 1.2.2.2

Simplify the left side.

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Step 1.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{| \frac{1}{3} \cdot (x + 3) |}{1} = \frac{0}{- 1}$

Step 1.2.2.2.2

Divide $| \frac{1}{3} \cdot (x + 3) |$ by $1$ .

$| \frac{1}{3} \cdot (x + 3) | = \frac{0}{- 1}$

Step 1.2.2.2.3

Apply the distributive property.

$| \frac{1}{3} x + \frac{1}{3} \cdot 3 | = \frac{0}{- 1}$

Step 1.2.2.2.4

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

$| \frac{x}{3} + \frac{1}{3} \cdot 3 | = \frac{0}{- 1}$

Step 1.2.2.2.5

Cancel the common factor of $3$ .

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Step 1.2.2.2.5.1

Cancel the common factor.

$| \frac{x}{3} + \frac{1}{3} \cdot 3 | = \frac{0}{- 1}$

Step 1.2.2.2.5.2

Rewrite the expression.

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

Step 1.2.2.3

Simplify the right side.

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Step 1.2.2.3.1

Divide $0$ by $- 1$ .

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

Step 1.2.3

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

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

Step 1.2.4

Plus or minus $0$ is $0$ .

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

Step 1.2.5

Subtract $1$ from both sides of the equation.

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

Step 1.2.6

Multiply both sides of the equation by $3$ .

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

Step 1.2.7

Simplify both sides of the equation.

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Step 1.2.7.1

Simplify the left side.

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Step 1.2.7.1.1

Cancel the common factor of $3$ .

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Step 1.2.7.1.1.1

Cancel the common factor.

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

Step 1.2.7.1.1.2

Rewrite the expression.

$x = 3 \cdot - 1$

Step 1.2.7.2

Simplify the right side.

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Step 1.2.7.2.1

Multiply $3$ by $- 1$ .

$x = - 3$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(- 3, 0)$

Step 2

Find the y-intercepts.

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Step 2.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = - | \frac{1}{3} \cdot ((0) + 3) |$

Step 2.2

Simplify $- | \frac{1}{3} \cdot ((0) + 3) |$ .

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Step 2.2.1

Add $0$ and $3$ .

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

Step 2.2.2

Cancel the common factor of $3$ .

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Step 2.2.2.1

Cancel the common factor.

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

Step 2.2.2.2

Rewrite the expression.

$y = - | 1 |$

Step 2.2.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$y = - 1 \cdot 1$

Step 2.2.4

Multiply $- 1$ by $1$ .

$y = - 1$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - 1)$

Step 3

List the intersections.

x-intercept(s): $(- 3, 0)$

y-intercept(s): $(0, - 1)$

Step 4