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Difficulty: MediumIntegers, Absolute Value, and Number Lines

What is the sum of all integer solutions to the equation ∣2x−8∣=14|2x - 8| = 14?

  1. A
    -3
  2. 8Answer
  3. C
    11
  4. D
    14
  5. E
    22

Answer

The sum of all integer solutions is 8.
To solve ∣2x−8∣=14|2x - 8| = 14, consider both possible cases: 2x−8=142x - 8 = 14 and 2x−8=−142x - 8 = -14. The first equation gives 2x=222x = 22, so x=11x = 11. The second equation gives 2x=−62x = -6, so x=−3x = -3. Adding these two solutions yields 11+(−3)=811 + (-3) = 8.

Step-by-Step Solution

1
Set up the two linear equations represented by the absolute value equation.
2x−8=142x - 8 = 14 or 2x−8=−142x - 8 = -14
The absolute value expression ∣x∣=k|x| = k for k>0k > 0 splits into x=kx = k and x=−kx = -k.
2
Solve the first equation for xx.
2x=22  ⟹  x=112x = 22 \implies x = 11
Add 8 to both sides and divide by 2.
3
Solve the second equation for xx.
2x=−6  ⟹  x=−32x = -6 \implies x = -3
Add 8 to both sides and divide by 2.
4
Calculate the sum of both integer solutions.
11+(−3)=811 + (-3) = 8
The problem asks for the sum of all integer solutions.

Key Concept

Solving Absolute Value Equations with Integers
Estimated Time:1m 0s
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Topics

Integers, Absolute Value, and Number Lines

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