How to Solve Absolute Value Equations With a Number on the Outside

By Grace Williams

Solving absolute value equations differs only slightly from solving linear equations. Absolute value equations are solved algebraically by isolating the variable, but such solutions require extra steps if there is a number outside the absolute value symbols.

Solve an absolute value equation that contains a number outside the absolute value bars by algebraically moving that number to the side of the equation opposite the variable. Eliminate the absolute value by creating two equations from the expression, representing the positive and negative possibilities for the terms within the bars. Solve for both answers.

Practice by solving the absolute value equation 2|x - 4| + 8 = 10 by first subtracting 8 from both sides: 2|x - 4| = 2. Divide both sides by 2: |x - 4| = 1. Eliminate the absolute value bars by writing two equations, to represent the positive and the negative possibilities of the interior subtraction: x - 4 = 1 and -(x - 4) = 1 or -x + 4 = 1.

Solve the equation x - 4 = 1 by adding 4 to both sides: x = 5. Solve the equation -x + 4 = 1 by subtracting 4 from both sides: -x = -3. Divide both sides by -1: x = 3. Write your final answer as x = 5 and x = 3.