Solve for a |ax+b|=c

$| a x + b | = c$

Step 1

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

$a x + b = \pm c$

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.

$a x + b = c$

Subtract $b$ from both sides of the equation.

$a x = c - b$

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

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

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

Simplify the left side.

Cancel the common factor of $x$ .

Cancel the common factor.

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

Divide $a$ by $1$ .

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

Simplify the right side.

Move the negative in front of the fraction.

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

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 $x$ and simplify.

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

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

Simplify the left side.

Cancel the common factor of $x$ .

Cancel the common factor.

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

Divide $a$ by $1$ .

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

Simplify the right side.

Simplify each term.

Move the negative in front of the fraction.

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

Move the negative in front of the fraction.

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

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

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

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

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

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

Solve for a |ax+b|=c

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