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

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

Step 1

Move all terms not containing $| 2 x - 3 |$ to the right side of the equation.

Subtract $2$ from both sides of the equation.

$3 | 2 x - 3 | = 3 - 2$

Subtract $2$ from $3$ .

$3 | 2 x - 3 | = 1$

Step 2

Divide each term in $3 | 2 x - 3 | = 1$ by $3$ and simplify.

Divide each term in $3 | 2 x - 3 | = 1$ by $3$ .

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

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

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

Divide $| 2 x - 3 |$ by $1$ .

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

Step 3

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

$2 x - 3 = \pm \frac{1}{3}$

Step 4

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.

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

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

Add $3$ to both sides of the equation.

$2 x = \frac{1}{3} + 3$

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $3$ by $3$ .

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

Add $1$ and $9$ .

$2 x = \frac{10}{3}$

Divide each term in $2 x = \frac{10}{3}$ by $2$ and simplify.

Divide each term in $2 x = \frac{10}{3}$ by $2$ .

$\frac{2 x}{2} = \frac{\frac{10}{3}}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{\frac{10}{3}}{2}$

Divide $x$ by $1$ .

$x = \frac{\frac{10}{3}}{2}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{10}{3} \cdot \frac{1}{2}$

Cancel the common factor of $2$ .

Factor $2$ out of $10$ .

$x = \frac{2 (5)}{3} \cdot \frac{1}{2}$

Cancel the common factor.

$x = \frac{2 \cdot 5}{3} \cdot \frac{1}{2}$

Rewrite the expression.

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

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

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

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

Add $3$ to both sides of the equation.

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $3$ by $3$ .

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

Add $- 1$ and $9$ .

$2 x = \frac{8}{3}$

Divide each term in $2 x = \frac{8}{3}$ by $2$ and simplify.

Divide each term in $2 x = \frac{8}{3}$ by $2$ .

$\frac{2 x}{2} = \frac{\frac{8}{3}}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{\frac{8}{3}}{2}$

Divide $x$ by $1$ .

$x = \frac{\frac{8}{3}}{2}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

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

Cancel the common factor of $2$ .

Factor $2$ out of $8$ .

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

Cancel the common factor.

$x = \frac{2 \cdot 4}{3} \cdot \frac{1}{2}$

Rewrite the expression.

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

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

$x = \frac{5}{3}, \frac{4}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{5}{3}, \frac{4}{3}$

Decimal Form:

$x = 1.\overline{6}, 1.\overline{3}$

Mixed Number Form:

$x = 1 \frac{2}{3}, 1 \frac{1}{3}$

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

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