Welcome to solving absolute value equations! An absolute value equation contains an expression inside absolute value bars. The absolute value represents the distance from zero on a number line, which is always positive or zero. Today we'll learn a systematic approach to solve these equations using the example |x - 2| = 3.
The first step in solving any absolute value equation is to isolate the absolute value expression. In our example |x - 2| = 3, the absolute value is already isolated on the left side. However, if we had a more complex equation like 2|x - 1| + 5 = 11, we would first subtract 5 from both sides to get 2|x - 1| = 6, then divide by 2 to get |x - 1| = 3.
Now we set up two separate equations. Since absolute value represents distance from zero, when |x - 2| equals 3, the expression x - 2 can be either positive 3 or negative 3. This gives us Case 1: x - 2 = 3, and Case 2: x - 2 = -3. The general rule is: if the absolute value of A equals B, then A equals B or A equals negative B.
Now we solve each equation separately. For Case 1, x - 2 = 3, we add 2 to both sides to get x = 5. For Case 2, x - 2 = -3, we add 2 to both sides to get x = -1. Therefore, our absolute value equation has two solutions: x = 5 and x = -1.
Finally, we check our solutions by substituting them back into the original equation. For x = 5: |5 - 2| = |3| = 3, which is correct. For x = -1: |-1 - 2| = |-3| = 3, which is also correct. On the number line, we can see that both -1 and 5 are exactly 3 units away from 2, confirming our solutions. This completes our method for solving absolute value equations.