Precalculus Examples

$f (x) = | 3 x + 5 | - 11$

Step 1

Set $| 3 x + 5 | - 11$ equal to $0$ .

$| 3 x + 5 | - 11 = 0$

Step 2

Solve for $x$ .

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Step 2.1

Add $11$ to both sides of the equation.

$| 3 x + 5 | = 11$

Step 2.2

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

$3 x + 5 = \pm 11$

Step 2.3

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

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Step 2.3.1

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

$3 x + 5 = 11$

Step 2.3.2

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

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Step 2.3.2.1

Subtract $5$ from both sides of the equation.

$3 x = 11 - 5$

Step 2.3.2.2

Subtract $5$ from $11$ .

$3 x = 6$

Step 2.3.3

Divide each term in $3 x = 6$ by $3$ and simplify.

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Step 2.3.3.1

Divide each term in $3 x = 6$ by $3$ .

$\frac{3 x}{3} = \frac{6}{3}$

Step 2.3.3.2

Simplify the left side.

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Step 2.3.3.2.1

Cancel the common factor of $3$ .

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Step 2.3.3.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{6}{3}$

Step 2.3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{6}{3}$

Step 2.3.3.3

Simplify the right side.

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Step 2.3.3.3.1

Divide $6$ by $3$ .

$x = 2$

Step 2.3.4

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

$3 x + 5 = - 11$

Step 2.3.5

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

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Step 2.3.5.1

Subtract $5$ from both sides of the equation.

$3 x = - 11 - 5$

Step 2.3.5.2

Subtract $5$ from $- 11$ .

$3 x = - 16$

Step 2.3.6

Divide each term in $3 x = - 16$ by $3$ and simplify.

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Step 2.3.6.1

Divide each term in $3 x = - 16$ by $3$ .

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

Step 2.3.6.2

Simplify the left side.

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Step 2.3.6.2.1

Cancel the common factor of $3$ .

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Step 2.3.6.2.1.1

Cancel the common factor.

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

Step 2.3.6.2.1.2

Divide $x$ by $1$ .

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

Step 2.3.6.3

Simplify the right side.

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Step 2.3.6.3.1

Move the negative in front of the fraction.

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

Step 2.3.7

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

$x = 2, - \frac{16}{3}$

Step 3