Solve these equations.
|x-7|=x-7

x ≥ 7

Step-by-step explanation:

|x - 7| = x - 7

A. For each absolute, find the intervals

x - 7 ≥ 0     x - 7 < 0

     x ≥ 7           x < 7

If x ≥ 7, |x - 7| = x - 7 > 0.

If x < 7, |x - 7| = x - 7 < 0. No solution.

B. Solve for x < 7

Rewrite |x - 7| = x - 7 as

+(x - 7) = x - 7

      x - 7 = x - 7

-x + 14 = x

        14 = 2x

          x = 7

7 ≮7. No solution

C. Solve for x ≥ 7

Rewrite |x - 7| = x - 7 as

+(x - 7) = x - 7

      x - 7 = x - 7

True for all x.

D. Merge overlapping intervals

No solution or x ≥ 7

⇒ x ≥ 7

The diagram below shows that the graphs of y = |x - 7| (blue) and of y = x - 7 (dashed red) coincide only when x ≥ 7.