Solve the Absolute Value Inequality for x 9|9-8x|=2x+3

$9 | 9 - 8 x | = 2 x + 3$

Step 1

Divide each term in $9 | 9 - 8 x | = 2 x + 3$ by $9$ and simplify.

Divide each term in $9 | 9 - 8 x | = 2 x + 3$ by $9$ .

$\frac{9 | 9 - 8 x |}{9} = \frac{2 x}{9} + \frac{3}{9}$

Simplify the left side.

Cancel the common factor of $9$ .

Cancel the common factor.

$\frac{9 | 9 - 8 x |}{9} = \frac{2 x}{9} + \frac{3}{9}$

Divide $| 9 - 8 x |$ by $1$ .

$| 9 - 8 x | = \frac{2 x}{9} + \frac{3}{9}$

Simplify the right side.

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

Factor $3$ out of $3$ .

$| 9 - 8 x | = \frac{2 x}{9} + \frac{3 (1)}{9}$

Cancel the common factors.

Factor $3$ out of $9$ .

$| 9 - 8 x | = \frac{2 x}{9} + \frac{3 \cdot 1}{3 \cdot 3}$

Cancel the common factor.

$| 9 - 8 x | = \frac{2 x}{9} + \frac{3 \cdot 1}{3 \cdot 3}$

Rewrite the expression.

$| 9 - 8 x | = \frac{2 x}{9} + \frac{1}{3}$

Step 2

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

$9 - 8 x = \pm (\frac{2 x}{9} + \frac{1}{3})$

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.

$9 - 8 x = \frac{2 x}{9} + \frac{1}{3}$

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

Subtract $\frac{2 x}{9}$ from both sides of the equation.

$9 - 8 x - \frac{2 x}{9} = \frac{1}{3}$

To write $- 8 x$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$9 - 8 x \cdot \frac{9}{9} - \frac{2 x}{9} = \frac{1}{3}$

Combine $- 8 x$ and $\frac{9}{9}$ .

$9 + \frac{- 8 x \cdot 9}{9} - \frac{2 x}{9} = \frac{1}{3}$

Combine the numerators over the common denominator.

$9 + \frac{- 8 x \cdot 9 - 2 x}{9} = \frac{1}{3}$

Simplify each term.

Simplify the numerator.

Factor $2 x$ out of $- 8 x \cdot 9 - 2 x$ .

Factor $2 x$ out of $- 8 x \cdot 9$ .

$9 + \frac{2 x (- 4 \cdot 9) - 2 x}{9} = \frac{1}{3}$

Factor $2 x$ out of $- 2 x$ .

$9 + \frac{2 x (- 4 \cdot 9) + 2 x (- 1)}{9} = \frac{1}{3}$

Factor $2 x$ out of $2 x (- 4 \cdot 9) + 2 x (- 1)$ .

$9 + \frac{2 x (- 4 \cdot 9 - 1)}{9} = \frac{1}{3}$

Multiply $- 4$ by $9$ .

$9 + \frac{2 x (- 36 - 1)}{9} = \frac{1}{3}$

Subtract $1$ from $- 36$ .

$9 + \frac{2 x \cdot - 37}{9} = \frac{1}{3}$

Multiply $- 37$ by $2$ .

$9 + \frac{- 74 x}{9} = \frac{1}{3}$

Move the negative in front of the fraction.

$9 - \frac{74 x}{9} = \frac{1}{3}$

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

Subtract $9$ from both sides of the equation.

$- \frac{74 x}{9} = \frac{1}{3} - 9$

To write $- 9$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{74 x}{9} = \frac{1}{3} - 9 \cdot \frac{3}{3}$

Combine $- 9$ and $\frac{3}{3}$ .

$- \frac{74 x}{9} = \frac{1}{3} + \frac{- 9 \cdot 3}{3}$

Combine the numerators over the common denominator.

$- \frac{74 x}{9} = \frac{1 - 9 \cdot 3}{3}$

Simplify the numerator.

Multiply $- 9$ by $3$ .

$- \frac{74 x}{9} = \frac{1 - 27}{3}$

Subtract $27$ from $1$ .

$- \frac{74 x}{9} = \frac{- 26}{3}$

Move the negative in front of the fraction.

$- \frac{74 x}{9} = - \frac{26}{3}$

Multiply both sides of the equation by $- \frac{9}{74}$ .

$- \frac{9}{74} (- \frac{74 x}{9}) = - \frac{9}{74} (- \frac{26}{3})$

Simplify both sides of the equation.

Simplify the left side.

Simplify $- \frac{9}{74} (- \frac{74 x}{9})$ .

Cancel the common factor of $9$ .

Move the leading negative in $- \frac{9}{74}$ into the numerator.

$\frac{- 9}{74} (- \frac{74 x}{9}) = - \frac{9}{74} (- \frac{26}{3})$

Move the leading negative in $- \frac{74 x}{9}$ into the numerator.

$\frac{- 9}{74} \cdot \frac{- 74 x}{9} = - \frac{9}{74} (- \frac{26}{3})$

Factor $9$ out of $- 9$ .

$\frac{9 (- 1)}{74} \cdot \frac{- 74 x}{9} = - \frac{9}{74} (- \frac{26}{3})$

Cancel the common factor.

$\frac{9 \cdot - 1}{74} \cdot \frac{- 74 x}{9} = - \frac{9}{74} (- \frac{26}{3})$

Rewrite the expression.

$\frac{- 1}{74} (- 74 x) = - \frac{9}{74} (- \frac{26}{3})$

Cancel the common factor of $74$ .

Factor $74$ out of $- 74 x$ .

$\frac{- 1}{74} (74 (- x)) = - \frac{9}{74} (- \frac{26}{3})$

Cancel the common factor.

$\frac{- 1}{74} (74 (- x)) = - \frac{9}{74} (- \frac{26}{3})$

Rewrite the expression.

$- - x = - \frac{9}{74} (- \frac{26}{3})$

Multiply.

Multiply $- 1$ by $- 1$ .

$1 x = - \frac{9}{74} (- \frac{26}{3})$

Multiply $x$ by $1$ .

$x = - \frac{9}{74} (- \frac{26}{3})$

Simplify the right side.

Simplify $- \frac{9}{74} (- \frac{26}{3})$ .

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{9}{74}$ into the numerator.

$x = \frac{- 9}{74} (- \frac{26}{3})$

Move the leading negative in $- \frac{26}{3}$ into the numerator.

$x = \frac{- 9}{74} \cdot \frac{- 26}{3}$

Factor $3$ out of $- 9$ .

$x = \frac{3 (- 3)}{74} \cdot \frac{- 26}{3}$

Cancel the common factor.

$x = \frac{3 \cdot - 3}{74} \cdot \frac{- 26}{3}$

Rewrite the expression.

$x = \frac{- 3}{74} \cdot - 26$

Cancel the common factor of $2$ .

Factor $2$ out of $74$ .

$x = \frac{- 3}{2 (37)} \cdot - 26$

Factor $2$ out of $- 26$ .

$x = \frac{- 3}{2 \cdot 37} \cdot (2 \cdot - 13)$

Cancel the common factor.

$x = \frac{- 3}{2 \cdot 37} \cdot (2 \cdot - 13)$

Rewrite the expression.

$x = \frac{- 3}{37} \cdot - 13$

Combine $\frac{- 3}{37}$ and $- 13$ .

$x = \frac{- 3 \cdot - 13}{37}$

Multiply $- 3$ by $- 13$ .

$x = \frac{39}{37}$

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

$9 - 8 x = - (\frac{2 x}{9} + \frac{1}{3})$

Simplify $- (\frac{2 x}{9} + \frac{1}{3})$ .

Rewrite.

$9 - 8 x = 0 + 0 - (\frac{2 x}{9} + \frac{1}{3})$

Simplify by adding zeros.

$9 - 8 x = - (\frac{2 x}{9} + \frac{1}{3})$

Apply the distributive property.

$9 - 8 x = - \frac{2 x}{9} - \frac{1}{3}$

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

Add $\frac{2 x}{9}$ to both sides of the equation.

$9 - 8 x + \frac{2 x}{9} = - \frac{1}{3}$

To write $- 8 x$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$9 - 8 x \cdot \frac{9}{9} + \frac{2 x}{9} = - \frac{1}{3}$

Combine $- 8 x$ and $\frac{9}{9}$ .

$9 + \frac{- 8 x \cdot 9}{9} + \frac{2 x}{9} = - \frac{1}{3}$

Combine the numerators over the common denominator.

$9 + \frac{- 8 x \cdot 9 + 2 x}{9} = - \frac{1}{3}$

Simplify each term.

Simplify the numerator.

Factor $2 x$ out of $- 8 x \cdot 9 + 2 x$ .

Factor $2 x$ out of $- 8 x \cdot 9$ .

$9 + \frac{2 x (- 4 \cdot 9) + 2 x}{9} = - \frac{1}{3}$

Factor $2 x$ out of $2 x$ .

$9 + \frac{2 x (- 4 \cdot 9) + 2 x (1)}{9} = - \frac{1}{3}$

Factor $2 x$ out of $2 x (- 4 \cdot 9) + 2 x (1)$ .

$9 + \frac{2 x (- 4 \cdot 9 + 1)}{9} = - \frac{1}{3}$

Multiply $- 4$ by $9$ .

$9 + \frac{2 x (- 36 + 1)}{9} = - \frac{1}{3}$

Add $- 36$ and $1$ .

$9 + \frac{2 x \cdot - 35}{9} = - \frac{1}{3}$

Multiply $- 35$ by $2$ .

$9 + \frac{- 70 x}{9} = - \frac{1}{3}$

Move the negative in front of the fraction.

$9 - \frac{70 x}{9} = - \frac{1}{3}$

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

Subtract $9$ from both sides of the equation.

$- \frac{70 x}{9} = - \frac{1}{3} - 9$

To write $- 9$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{70 x}{9} = - \frac{1}{3} - 9 \cdot \frac{3}{3}$

Combine $- 9$ and $\frac{3}{3}$ .

$- \frac{70 x}{9} = - \frac{1}{3} + \frac{- 9 \cdot 3}{3}$

Combine the numerators over the common denominator.

$- \frac{70 x}{9} = \frac{- 1 - 9 \cdot 3}{3}$

Simplify the numerator.

Multiply $- 9$ by $3$ .

$- \frac{70 x}{9} = \frac{- 1 - 27}{3}$

Subtract $27$ from $- 1$ .

$- \frac{70 x}{9} = \frac{- 28}{3}$

Move the negative in front of the fraction.

$- \frac{70 x}{9} = - \frac{28}{3}$

Multiply both sides of the equation by $- \frac{9}{70}$ .

$- \frac{9}{70} (- \frac{70 x}{9}) = - \frac{9}{70} (- \frac{28}{3})$

Simplify both sides of the equation.

Simplify the left side.

Simplify $- \frac{9}{70} (- \frac{70 x}{9})$ .

Cancel the common factor of $9$ .

Move the leading negative in $- \frac{9}{70}$ into the numerator.

$\frac{- 9}{70} (- \frac{70 x}{9}) = - \frac{9}{70} (- \frac{28}{3})$

Move the leading negative in $- \frac{70 x}{9}$ into the numerator.

$\frac{- 9}{70} \cdot \frac{- 70 x}{9} = - \frac{9}{70} (- \frac{28}{3})$

Factor $9$ out of $- 9$ .

$\frac{9 (- 1)}{70} \cdot \frac{- 70 x}{9} = - \frac{9}{70} (- \frac{28}{3})$

Cancel the common factor.

$\frac{9 \cdot - 1}{70} \cdot \frac{- 70 x}{9} = - \frac{9}{70} (- \frac{28}{3})$

Rewrite the expression.

$\frac{- 1}{70} (- 70 x) = - \frac{9}{70} (- \frac{28}{3})$

Cancel the common factor of $70$ .

Factor $70$ out of $- 70 x$ .

$\frac{- 1}{70} (70 (- x)) = - \frac{9}{70} (- \frac{28}{3})$

Cancel the common factor.

$\frac{- 1}{70} (70 (- x)) = - \frac{9}{70} (- \frac{28}{3})$

Rewrite the expression.

$- - x = - \frac{9}{70} (- \frac{28}{3})$

Multiply.

Multiply $- 1$ by $- 1$ .

$1 x = - \frac{9}{70} (- \frac{28}{3})$

Multiply $x$ by $1$ .

$x = - \frac{9}{70} (- \frac{28}{3})$

Simplify the right side.

Simplify $- \frac{9}{70} (- \frac{28}{3})$ .

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{9}{70}$ into the numerator.

$x = \frac{- 9}{70} (- \frac{28}{3})$

Move the leading negative in $- \frac{28}{3}$ into the numerator.

$x = \frac{- 9}{70} \cdot \frac{- 28}{3}$

Factor $3$ out of $- 9$ .

$x = \frac{3 (- 3)}{70} \cdot \frac{- 28}{3}$

Cancel the common factor.

$x = \frac{3 \cdot - 3}{70} \cdot \frac{- 28}{3}$

Rewrite the expression.

$x = \frac{- 3}{70} \cdot - 28$

Cancel the common factor of $14$ .

Factor $14$ out of $70$ .

$x = \frac{- 3}{14 (5)} \cdot - 28$

Factor $14$ out of $- 28$ .

$x = \frac{- 3}{14 \cdot 5} \cdot (14 \cdot - 2)$

Cancel the common factor.

$x = \frac{- 3}{14 \cdot 5} \cdot (14 \cdot - 2)$

Rewrite the expression.

$x = \frac{- 3}{5} \cdot - 2$

Combine $\frac{- 3}{5}$ and $- 2$ .

$x = \frac{- 3 \cdot - 2}{5}$

Multiply $- 3$ by $- 2$ .

$x = \frac{6}{5}$

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

$x = \frac{39}{37}, \frac{6}{5}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{39}{37}, \frac{6}{5}$

Decimal Form:

$x = 1.\overline{054}, 1.2$

Mixed Number Form:

$x = 1 \frac{2}{37}, 1 \frac{1}{5}$

Solve the Absolute Value Inequality for x 9|9-8x|=2x+3

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