16Hit1600
AlgebraHardAbsolute value inequalities

Algebra practice question

Which of the following is the complete solution set to |x − 4| < 6?

  1. A. x < 10
  2. B. −2 < x < 10Correct
  3. C. x < −2 or x > 10
  4. D. 2 < x < 10

Answer: B. −2 < x < 10

A 'less than' absolute value becomes a compound AND inequality: −6 < x − 4 < 6. Add 4 to all parts: −2 < x < 10. Choice C is the 'greater than' (OR) form, which applies to |x − 4| > 6, not this one.

Why the other answers are wrong

A. x < 10
You solved only x − 4 < 6. A 'less than' absolute value also requires x − 4 > −6, which adds the lower bound x > −2.
C. x < −2 or x > 10
That's the OR form, which solves |x − 4| > 6. Since this inequality is 'less than', the solution is the single interval −2 < x < 10.
D. 2 < x < 10
Your upper bound is right, but on the lower side you computed 6 − 4 = 2 instead of −6 + 4 = −2. Adding 4 to −6 gives x > −2.

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