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Difficulty: EasyInequalities and Absolute Value Equations

If ∣3−2x∣=7|3 - 2x| = 7, what is the product of all possible real values of xx?

  1. −10-10Answer
  2. B
    −3-3
  3. C
    33
  4. D
    55
  5. E
    1010

Answer

The product of all possible real values of xx is −10-10.
Solving ∣3−2x∣=7|3 - 2x| = 7 yields two equations: 3−2x=73 - 2x = 7, which gives x=−2x = -2, and 3−2x=−73 - 2x = -7, which gives x=5x = 5. Multiplying these two values together gives (−2)×5=−10(-2) \times 5 = -10.

Step-by-Step Solution

1
Set up two linear equations corresponding to the positive and negative cases of the absolute value expression.
3−2x=73 - 2x = 7 or 3−2x=−73 - 2x = -7
By definition, ∣a∣=b|a| = b (where b≥0b \ge 0) implies a=ba = b or a=−ba = -b.
2
Solve the first linear equation for xx.
−2x=4  ⟹  x=−2-2x = 4 \implies x = -2
Subtract 3 from both sides and divide by −2-2.
3
Solve the second linear equation for xx.
−2x=−10  ⟹  x=5-2x = -10 \implies x = 5
Subtract 3 from both sides and divide by −2-2.
4
Calculate the product of the two solutions.
(−2)×5=−10(-2) \times 5 = -10
The question asks for the product of all possible real values of xx.

Key Concept

Absolute Value Equations
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Topics

Inequalities and Absolute Value Equations

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