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Math · 06 · Algebra

Inequalities and Absolute Value

Notes

Key points

  • Multiply or divide by a negative → flip the inequality.
  • Absolute value = distance: ∣x−a∣|x-a| is the distance from xx to aa.
  • “Less than” → between; “greater than” → outside.
  • Absolute value equal to a negative → no solution.
  • “At least” ≥, “at most” / “no more than” ≤, “fewer than” <.
  • Graphs: solid boundary for ≤ ≥, dashed for < >. Solution = overlap of shadings.

Formulas

Absolute value equation∣x−a∣=b ⇒ x=a±b|x-a|=b\ \Rightarrow\ x=a\pm bOnly when b ≥ 0
Between∣x−a∣<b  ⟺  a−b<x<a+b|x-a|<b\iff a-b<x<a+b
Outside∣x−a∣>b  ⟺  x<a−b or x>a+b|x-a|>b\iff x<a-b\ \text{or}\ x>a+b
Range to absolute valueL≤x≤H  ⟺  ∣x−L+H2∣≤H−L2L\le x\le H\iff\left|x-\dfrac{L+H}{2}\right|\le\dfrac{H-L}{2}Centre and half-width

Common traps

  • Forgetting to flip the sign after dividing by a negative.
  • Maximum whole items: round down (13.8→1313.8\to13), even past .5.
  • Testing a point that sits on a dashed boundary — it isn't a solution.

SAT tip

“Which point is a solution to the system?” Plug each choice into both inequalities, or graph both and check the overlap.

Test yourself

10 official SAT questions on Inequalities and Absolute Value, easiest first.

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