Piecewise Behavior Math Example 2

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Example 2

medium
Solve the equation ∣2x−5∣=7|2x - 5| = 7 and the inequality ∣2x−5∣<7|2x - 5| < 7.

Solution

  1. 1
    Equation: ∣2x−5∣=7|2x-5|=7 means 2x−5=72x-5=7 or 2x−5=−72x-5=-7. Case 1: 2x=12⇒x=62x=12 \Rightarrow x=6. Case 2: 2x=−2⇒x=−12x=-2 \Rightarrow x=-1.
  2. 2
    Inequality: ∣2x−5∣<7|2x-5|<7 means −7<2x−5<7⇒−2<2x<12⇒−1<x<6-7 < 2x-5 < 7 \Rightarrow -2 < 2x < 12 \Rightarrow -1 < x < 6.
  3. 3
    So the solution set of the inequality is the open interval (−1,6)(-1, 6).

Answer

Equation: x=6x=6 or x=−1x=-1; Inequality: −1<x<6-1 < x < 6
Absolute value equations split into two cases (positive/negative branch). Absolute value inequalities of the form ∣u∣<a|u|<a convert to the compound inequality −a<u<a-a<u<a.

About Piecewise Behavior

Piecewise behavior refers to a function that exhibits qualitatively different characteristics in different regions of its domain, like having a different slope or curvature in each region.

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More Piecewise Behavior Examples

Example 1 easy

Write [formula] as an explicit piecewise function, evaluate [formula], [formula], and [formula], and

Example 3 easy

Simplify [formula] for [formula], [formula], and [formula].

Example 4 hard

Express [formula] as a piecewise function (without absolute value bars) and identify where [formula]

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