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Difficulty: EasyInequalities and Absolute Value Equations

If ∣2x−5∣=9|2x - 5| = 9, what is the sum of all possible values of xx?

  1. 55Answer
  2. B
    77
  3. C
    −2-2
  4. D
    99
  5. E
    1414

Answer

The sum of all possible values of xx is 55.
To solve ∣2x−5∣=9|2x - 5| = 9, split the equation into two linear cases: 2x−5=92x - 5 = 9 and 2x−5=−92x - 5 = -9. Solving 2x−5=92x - 5 = 9 gives 2x=142x = 14, so x=7x = 7. Solving 2x−5=−92x - 5 = -9 gives 2x=−42x = -4, so x=−2x = -2. Summing these two solutions gives 7+(−2)=57 + (-2) = 5.

Step-by-Step Solution

1
Set up the two equations based on the definition of absolute value.
2x−5=92x - 5 = 9 and 2x−5=−92x - 5 = -9
For any expression AA and real number B≥0B \ge 0, ∣A∣=B|A| = B implies A=BA = B or A=−BA = -B.
2
Solve the first equation for xx.
2x=14  ⟹  x=72x = 14 \implies x = 7
Add 55 to both sides and divide by 22.
3
Solve the second equation for xx.
2x=−4  ⟹  x=−22x = -4 \implies x = -2
Add 55 to both sides and divide by 22.
4
Calculate the sum of all valid solutions.
7+(−2)=57 + (-2) = 5
The question asks for the sum of all possible solutions for xx.

Key Concept

Solving Absolute Value Equations
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Topics

Inequalities and Absolute Value Equations

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