Solve the Absolute Value Inequality for x |2x-3|=4x-1

$| 2 x - 3 | = 4 x - 1$

Step 1

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

$2 x - 3 = \pm (4 x - 1)$

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.

$2 x - 3 = 4 x - 1$

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

Subtract $4 x$ from both sides of the equation.

$2 x - 3 - 4 x = - 1$

Subtract $4 x$ from $2 x$ .

$- 2 x - 3 = - 1$

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

Add $3$ to both sides of the equation.

$- 2 x = - 1 + 3$

Add $- 1$ and $3$ .

$- 2 x = 2$

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

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

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

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Divide $2$ by $- 2$ .

$x = - 1$

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

$2 x - 3 = - (4 x - 1)$

Simplify $- (4 x - 1)$ .

Rewrite.

$2 x - 3 = 0 + 0 - (4 x - 1)$

Simplify by adding zeros.

$2 x - 3 = - (4 x - 1)$

Apply the distributive property.

$2 x - 3 = - (4 x) - - 1$

Multiply.

Multiply $4$ by $- 1$ .

$2 x - 3 = - 4 x - - 1$

Multiply $- 1$ by $- 1$ .

$2 x - 3 = - 4 x + 1$

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

Add $4 x$ to both sides of the equation.

$2 x - 3 + 4 x = 1$

Add $2 x$ and $4 x$ .

$6 x - 3 = 1$

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

Add $3$ to both sides of the equation.

$6 x = 1 + 3$

Add $1$ and $3$ .

$6 x = 4$

Divide each term in $6 x = 4$ by $6$ and simplify.

Divide each term in $6 x = 4$ by $6$ .

$\frac{6 x}{6} = \frac{4}{6}$

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6 x}{6} = \frac{4}{6}$

Divide $x$ by $1$ .

$x = \frac{4}{6}$

Simplify the right side.

Cancel the common factor of $4$ and $6$ .

Factor $2$ out of $4$ .

$x = \frac{2 (2)}{6}$

Cancel the common factors.

Factor $2$ out of $6$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

Step 3

Exclude the solutions that do not make $| 2 x - 3 | = 4 x - 1$ true.

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

Step 4

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 0.\overline{6}$

Solve the Absolute Value Inequality for x |2x-3|=4x-1

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