Absolute Value Equation

[nycmath]

Solve | x - 5 | = 7 for x.

To solve the absolute value equation | x - 5 | = 7, we must consider two cases because the absolute value of a number represents its distance from zero on a number line.

Case 1

x - 5 = 7

x = 7 + 5

x = 12

Case 2

x - 5 = -7

x = -7 + 5

x = -2

We must now check both values of x found to determine which one satisfies the original absolute value equation.

Let x = 12

| x - 5 | = 7

| 12 - 5 | = 7

| 7 | = 7

7 = 7

This is true. We now know that one of values of x is 12.

Let x = -2

| x - 5 | = 7

| -2 - 5 | = 7

| - 7 | = 7

7 = 7

This is also true.

Answer: x = - 2 and x = 12.

You try one.

Solve for x.

| x + 6 | = 12