Solving Absolute Value Equations

Definition: The absolute value of a number x is the distance from x to zero on a number line, or |x|. The absolute value of a number is always positive.

When solving absolute value equations, we must consider both the negative solution and positive solution. For example, given |x| = 5, there are two solutions for which the value of x is 5 units away from zero:

|5| = 5 or |-5| = 5,

so x = 5 or x = -5.

This is the same as rewriting |x| = 5 as x = ±5.

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Example: Solve the equation for x: |x + 3| = 8

Solution: 

Step 1: Rewrite the equation keeping in mind that |x + 3| = 8 is the same as x + 3 = ±8

Step 2: Our two new equations are
x + 3 = 8
x + 3 = -8

Step 3: Solve each equation to find the two solutions:

x + 3 = 8
x = 8 – 3
x = 5

x + 3 = -8
x = -8 – 3
x = -11

Step 4: Plug both solutions back into the original equation to verify:

|x + 3| = 8
|5 + 3| = 8
|8| = 8

|x +3| = 8
|-11 + 3| = 8
|-8| = 8

Both equations hold true, so our solutions are x = 5, x = -11.

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