Question

The sum of the possible values of \(x\) in the equation \(|x+7| + |x-8| = 16\) is:

• A. 0
• B. 1
• C. 2
• D. 3
• E. None of the above

Hint:

When solving absolute value equations, always break into intervals and check feasibility of each solution.

Answer 1

The Correct Option is B

We solve the absolute value equation by considering different intervals of \(x\):
Step 1: Case 1 (\(x \geq 8\)) \[ |x+7| = x+7, |x-8| = x-8 \] \[ (x+7) + (x-8) = 16 ⇒ 2x - 1 = 16 ⇒ 2x = 17 ⇒ x = \tfrac{17}{2} = 8.5 \] This satisfies \(x \geq 8\).
Step 2: Case 2 (\(-7 \leq x<8\)) \[ |x+7| = x+7, |x-8| = -(x-8) = -x+8 \] \[ (x+7) + (-x+8) = 16 ⇒ 15 = 16 \] Contradiction. No solution.
Step 3: Case 3 (\(x<-7\)) \[ |x+7| = -(x+7) = -x-7, |x-8| = -(x-8) = -x+8 \] \[ (-x-7) + (-x+8) = 16 ⇒ -2x+1 = 16 ⇒ -2x = 15 ⇒ x = -7.5 \] This satisfies \(x<-7\).
Step 4: Sum of solutions \[ x = 8.5 \text{and} x = -7.5 \] \[ \text{Sum} = 8.5 + (-7.5) = 1 \] \[ \boxed{1} \]