AlgebraLab

Solving Absolute Value Equations

By S Taylor

Think about the equation |x| = 3. This means that x could be 3 or x could be -3. When you take the absolute value of 3, the answer is 3 and when you take the absolute value of -3, the answer is also 3. So when you have an absolute value equation, you have to take into account that there can be two answers that will make the equation true.

In general, to solve an absolute value equation, you set the quantity inside the absolute value sign equal to the positive and negative value on the other side of the equal sign.

To solve |expression| = k, you must solve two separate equations:

and


Let's Practice
  1. |x + 1| = 4
    2. The quantity inside the absolute value sign can be equal to 4 or -4.
    and
    So the solutions are and .
  2. |2x - 3| = x - 5
    3. When solving this equation, you have to be careful when working with the opposite of .

    and
    So the solutions are and .
  3. |x + 6| = -2
    4. Before jumping into solving this problem, think about it first. This equation is saying that the absolute value of some quantity is equal to -2. But the absolute value of a number is always positive. So it is impossible for the answer to be a negative number. There is no solution to this equation.
  4. |x - 10| = x2 - 10x
    5. Although solving this equation involves the same process as used before, it will be a little more involved since it involves a quadratic. If you need to review how to solve a quadratic equation, click here to go to the solving quadratics lesson.

    and
    So the solutions are , , and .
1.
|x + 7| = 8
2.
|3x - 4| = 10
3.
|2x + 1| = x - 6
4.
|7x - 9| = -12
5.
|x - 1| = x2