Algebra Examples

$f (x) = 3 | x + 4 | - 4$

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 $x$ .

$0 = 3 | x + 4 | - 4$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $3 | x + 4 | - 4 = 0$ .

$3 | x + 4 | - 4 = 0$

Step 1.2.2

Add $4$ to both sides of the equation.

$3 | x + 4 | = 4$

Step 1.2.3

Divide each term in $3 | x + 4 | = 4$ by $3$ and simplify.

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Step 1.2.3.1

Divide each term in $3 | x + 4 | = 4$ by $3$ .

$\frac{3 | x + 4 |}{3} = \frac{4}{3}$

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

Cancel the common factor of $3$ .

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Step 1.2.3.2.1.1

Cancel the common factor.

$\frac{3 | x + 4 |}{3} = \frac{4}{3}$

Step 1.2.3.2.1.2

Divide $| x + 4 |$ by $1$ .

$| x + 4 | = \frac{4}{3}$

Step 1.2.4

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

$x + 4 = \pm \frac{4}{3}$

Step 1.2.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 1.2.5.1

First, use the positive value of the $\pm$ to find the first solution.

$x + 4 = \frac{4}{3}$

Step 1.2.5.2

Move all terms not containing $x$ to the right side of the equation.

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Step 1.2.5.2.1

Subtract $4$ from both sides of the equation.

$x = \frac{4}{3} - 4$

Step 1.2.5.2.2

To write $- 4$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x = \frac{4}{3} - 4 \cdot \frac{3}{3}$

Step 1.2.5.2.3

Combine $- 4$ and $\frac{3}{3}$ .

$x = \frac{4}{3} + \frac{- 4 \cdot 3}{3}$

Step 1.2.5.2.4

Combine the numerators over the common denominator.

$x = \frac{4 - 4 \cdot 3}{3}$

Step 1.2.5.2.5

Simplify the numerator.

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Step 1.2.5.2.5.1

Multiply $- 4$ by $3$ .

$x = \frac{4 - 12}{3}$

Step 1.2.5.2.5.2

Subtract $12$ from $4$ .

$x = \frac{- 8}{3}$

Step 1.2.5.2.6

Move the negative in front of the fraction.

$x = - \frac{8}{3}$

Step 1.2.5.3

Next, use the negative value of the $\pm$ to find the second solution.

$x + 4 = - \frac{4}{3}$

Step 1.2.5.4

Move all terms not containing $x$ to the right side of the equation.

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Step 1.2.5.4.1

Subtract $4$ from both sides of the equation.

$x = - \frac{4}{3} - 4$

Step 1.2.5.4.2

To write $- 4$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x = - \frac{4}{3} - 4 \cdot \frac{3}{3}$

Step 1.2.5.4.3

Combine $- 4$ and $\frac{3}{3}$ .

$x = - \frac{4}{3} + \frac{- 4 \cdot 3}{3}$

Step 1.2.5.4.4

Combine the numerators over the common denominator.

$x = \frac{- 4 - 4 \cdot 3}{3}$

Step 1.2.5.4.5

Simplify the numerator.

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Step 1.2.5.4.5.1

Multiply $- 4$ by $3$ .

$x = \frac{- 4 - 12}{3}$

Step 1.2.5.4.5.2

Subtract $12$ from $- 4$ .

$x = \frac{- 16}{3}$

Step 1.2.5.4.6

Move the negative in front of the fraction.

$x = - \frac{16}{3}$

Step 1.2.5.5

The complete solution is the result of both the positive and negative portions of the solution.

$x = - \frac{8}{3}, - \frac{16}{3}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(- \frac{8}{3}, 0), (- \frac{16}{3}, 0)$

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 $y$ .

$y = 3 | (0) + 4 | - 4$

Step 2.2

Simplify $3 | (0) + 4 | - 4$ .

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Add $0$ and $4$ .

$y = 3 | 4 | - 4$

Step 2.2.1.2

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

$y = 3 \cdot 4 - 4$

Step 2.2.1.3

Multiply $3$ by $4$ .

$y = 12 - 4$

Step 2.2.2

Subtract $4$ from $12$ .

$y = 8$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 8)$

Step 3

List the intersections.

x-intercept(s): $(- \frac{8}{3}, 0), (- \frac{16}{3}, 0)$

y-intercept(s): $(0, 8)$

Step 4