Solve the Absolute Value Inequality for x 3|2x+5|=9x-6

$3 | 2 x + 5 | = 9 x - 6$

Step 1

Divide each term in $3 | 2 x + 5 | = 9 x - 6$ by $3$ and simplify.

Divide each term in $3 | 2 x + 5 | = 9 x - 6$ by $3$ .

$\frac{3 | 2 x + 5 |}{3} = \frac{9 x}{3} + \frac{- 6}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 | 2 x + 5 |}{3} = \frac{9 x}{3} + \frac{- 6}{3}$

Divide $| 2 x + 5 |$ by $1$ .

$| 2 x + 5 | = \frac{9 x}{3} + \frac{- 6}{3}$

Simplify the right side.

Simplify each term.

Cancel the common factor of $9$ and $3$ .

Factor $3$ out of $9 x$ .

$| 2 x + 5 | = \frac{3 (3 x)}{3} + \frac{- 6}{3}$

Cancel the common factors.

Factor $3$ out of $3$ .

$| 2 x + 5 | = \frac{3 (3 x)}{3 (1)} + \frac{- 6}{3}$

Cancel the common factor.

$| 2 x + 5 | = \frac{3 (3 x)}{3 \cdot 1} + \frac{- 6}{3}$

Rewrite the expression.

$| 2 x + 5 | = \frac{3 x}{1} + \frac{- 6}{3}$

Divide $3 x$ by $1$ .

$| 2 x + 5 | = 3 x + \frac{- 6}{3}$

Divide $- 6$ by $3$ .

$| 2 x + 5 | = 3 x - 2$

Step 2

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

$2 x + 5 = \pm (3 x - 2)$

Step 3

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.

$2 x + 5 = 3 x - 2$

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

Subtract $3 x$ from both sides of the equation.

$2 x + 5 - 3 x = - 2$

Subtract $3 x$ from $2 x$ .

$- x + 5 = - 2$

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

Subtract $5$ from both sides of the equation.

$- x = - 2 - 5$

Subtract $5$ from $- 2$ .

$- x = - 7$

Divide each term in $- x = - 7$ by $- 1$ and simplify.

Divide each term in $- x = - 7$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 7}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 7}{- 1}$

Divide $x$ by $1$ .

$x = \frac{- 7}{- 1}$

Simplify the right side.

Divide $- 7$ by $- 1$ .

$x = 7$

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

$2 x + 5 = - (3 x - 2)$

Simplify $- (3 x - 2)$ .

Rewrite.

$2 x + 5 = 0 + 0 - (3 x - 2)$

Simplify by adding zeros.

$2 x + 5 = - (3 x - 2)$

Apply the distributive property.

$2 x + 5 = - (3 x) - - 2$

Multiply.

Multiply $3$ by $- 1$ .

$2 x + 5 = - 3 x - - 2$

Multiply $- 1$ by $- 2$ .

$2 x + 5 = - 3 x + 2$

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

Add $3 x$ to both sides of the equation.

$2 x + 5 + 3 x = 2$

Add $2 x$ and $3 x$ .

$5 x + 5 = 2$

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

Subtract $5$ from both sides of the equation.

$5 x = 2 - 5$

Subtract $5$ from $2$ .

$5 x = - 3$

Divide each term in $5 x = - 3$ by $5$ and simplify.

Divide each term in $5 x = - 3$ by $5$ .

$\frac{5 x}{5} = \frac{- 3}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 x}{5} = \frac{- 3}{5}$

Divide $x$ by $1$ .

$x = \frac{- 3}{5}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{3}{5}$

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

$x = 7, - \frac{3}{5}$

Step 4

Exclude the solutions that do not make $3 | 2 x + 5 | = 9 x - 6$ true.

$x = 7$

Solve the Absolute Value Inequality for x 3|2x+5|=9x-6

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