|u| + 11 < 22
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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.
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