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 < -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 > 1 or x < -1) and x <−1.x < -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).