How to solve this absolute value inequality

label Algebra
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Solve.

|u| + 11 < 22

Sep 14th, 2015

Thank you for the opportunity to help you with your question!

Hi, Rachel

Step 1: Isolate the variable: that is, make it stand alone on one side of the '<' sign.

Subtract 11 from both sides: |u| + 11 - 11 < 22 - 11.
Simplify: |u| < 11.

Step 2. Recall the definition of absolute value: it's a distance (which is always positive or zero).

if (0 <= x) then x = x, otherwise x = -x.

Step 3. Rewrite |u| < 11 for: a) when u < 0, and; b) when 0 <= u.

a) If u < 0, then |u| = -u, so we have -u < 11, but -u < 11 only if u > -11

b) If 0 <= u, then |u| = u, so we have u < 11.

Step 4. Combine results from a) and b) above.

-11 < u < 11.

CHECK: Suppose u = 10. Then is |10| + 11 < 22?  YES

CHECK: Suppose u = -9. Then  is |-9| + 11 < 22? Yes, since |-9| = 9.

  

Please let me know if you need any clarification. I'm always happy to answer your questions.
Sep 14th, 2015

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