Solve for x |ax+b|=c

$| a x + b | = c$

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

$a x + b = \pm c$

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.

$a x + b = c$

Subtract $b$ from both sides of the equation.

$a x = c - b$

Divide each term in $a x = c - b$ by $a$ and simplify.

Divide each term in $a x = c - b$ by $a$ .

$\frac{a x}{a} = \frac{c}{a} + \frac{- b}{a}$

Simplify the left side.

Cancel the common factor of $a$ .

Cancel the common factor.

$\frac{a x}{a} = \frac{c}{a} + \frac{- b}{a}$

Divide $x$ by $1$ .

$x = \frac{c}{a} + \frac{- b}{a}$

Simplify the right side.

Move the negative in front of the fraction.

$x = \frac{c}{a} - \frac{b}{a}$

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

$a x + b = - c$

Subtract $b$ from both sides of the equation.

$a x = - c - b$

Divide each term in $a x = - c - b$ by $a$ and simplify.

Divide each term in $a x = - c - b$ by $a$ .

$\frac{a x}{a} = \frac{- c}{a} + \frac{- b}{a}$

Simplify the left side.

Cancel the common factor of $a$ .

Cancel the common factor.

$\frac{a x}{a} = \frac{- c}{a} + \frac{- b}{a}$

Divide $x$ by $1$ .

$x = \frac{- c}{a} + \frac{- b}{a}$

Simplify the right side.

Simplify each term.

Move the negative in front of the fraction.

$x = - \frac{c}{a} + \frac{- b}{a}$

Move the negative in front of the fraction.

$x = - \frac{c}{a} - \frac{b}{a}$

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

$x = \frac{c}{a} - \frac{b}{a}$

$x = - \frac{c}{a} - \frac{b}{a}$

$x = \frac{c}{a} - \frac{b}{a}$

$x = - \frac{c}{a} - \frac{b}{a}$