Solve the Absolute Value Inequality for x |4x-1|=2x+13

$| 4 x - 1 | = 2 x + 13$

Step 1

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

$4 x - 1 = \pm (2 x + 13)$

Step 2

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

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

$4 x - 1 = 2 x + 13$

Move all terms containing $x$ to the left side of the equation.

Subtract $2 x$ from both sides of the equation.

$4 x - 1 - 2 x = 13$

Subtract $2 x$ from $4 x$ .

$2 x - 1 = 13$

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

Add $1$ to both sides of the equation.

$2 x = 13 + 1$

Add $13$ and $1$ .

$2 x = 14$

Divide each term in $2 x = 14$ by $2$ and simplify.

Divide each term in $2 x = 14$ by $2$ .

$\frac{2 x}{2} = \frac{14}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{14}{2}$

Divide $x$ by $1$ .

$x = \frac{14}{2}$

Simplify the right side.

Divide $14$ by $2$ .

$x = 7$

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

$4 x - 1 = - (2 x + 13)$

Simplify $- (2 x + 13)$ .

Rewrite.

$4 x - 1 = 0 + 0 - (2 x + 13)$

Simplify by adding zeros.

$4 x - 1 = - (2 x + 13)$

Apply the distributive property.

$4 x - 1 = - (2 x) - 1 \cdot 13$

Multiply.

Multiply $2$ by $- 1$ .

$4 x - 1 = - 2 x - 1 \cdot 13$

Multiply $- 1$ by $13$ .

$4 x - 1 = - 2 x - 13$

Move all terms containing $x$ to the left side of the equation.

Add $2 x$ to both sides of the equation.

$4 x - 1 + 2 x = - 13$

Add $4 x$ and $2 x$ .

$6 x - 1 = - 13$

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

Add $1$ to both sides of the equation.

$6 x = - 13 + 1$

Add $- 13$ and $1$ .

$6 x = - 12$

Divide each term in $6 x = - 12$ by $6$ and simplify.

Divide each term in $6 x = - 12$ by $6$ .

$\frac{6 x}{6} = \frac{- 12}{6}$

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6 x}{6} = \frac{- 12}{6}$

Divide $x$ by $1$ .

$x = \frac{- 12}{6}$

Simplify the right side.

Divide $- 12$ by $6$ .

$x = - 2$

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

$x = 7, - 2$

Solve the Absolute Value Inequality for x |4x-1|=2x+13

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