2211.21 – Product with an Absolute Value

If xx is a real number, then for what values of xx is (1−∣x∣)(1+ x)(1 - |x|)(1 + x) a positive number?

  1. x<1x < 1

  2. x=1x = 1

  3. x>1x > 1

  4. x<−1x < -1

  5. x<−1x < -1 or −1<x<1-1 < x < 1

Solution

The quantity (1−∣x∣)(1+ x)(1 - |x|)(1 + x) is positive if both factors are positive OR if both factors are negative. We must consider both cases.

  • Case 1: 1−∣x∣>01 - |x| > 0 and 1+x>01 + x > 0. This means 1>∣x∣1 > |x| and x>−1x > -1, or −1<x<1-1 < x < 1 and x>−1x > -1. The conclusion in this case is −1<x<1-1 < x < 1.

  • Case 2: 1−∣x∣<01 - |x| < 0 and 1+x<01 + x < 0. This means ∣x∣>1|x| > 1 and x<−1x < -1 , which yields (x>1x > 1 or x<−1x < -1) and x<−1x < -1. The conclusion in this case is x<−1x < -1.

Together the two cases yield x<−1x < -1 or −1<x<1-1 < x < 1. The answer is (e).