Inequalities and Absolute Value
Extending equations to ranges — solving inequalities, understanding absolute value, and working with intervals.
Inequalities
An inequality is a relation between two expressions using <, >, ≤, or ≥ instead of =. Where an equation has specific solutions, an inequality has a range.
x + 3 > 7 → x > 4 (all numbers greater than 4)
2x ≤ 10 → x ≤ 5 (all numbers up to and including 5)
Solving Linear Inequalities
Same rules as equations — with one critical difference:
Multiplying or dividing by a negative number reverses the inequality sign.
−2x < 6
x > −3 (sign flips when dividing by −2)
Why: multiplying by a negative reflects the number line. What was “greater” is now “lesser.”
3x − 4 > 11
3x > 15
x > 5
−3x + 2 ≤ 14
−3x ≤ 12
x ≥ −4 (flip the sign)
Interval Notation
Inequalities are written as intervals — a compact notation for ranges of values.
| Inequality | Interval | Meaning |
|---|---|---|
| x > 4 | (4, ∞) | open at 4, not included |
| x ≥ 4 | [4, ∞) | closed at 4, included |
| x < 4 | (−∞, 4) | open at 4 |
| x ≤ 4 | (−∞, 4] | closed at 4 |
| 2 < x ≤ 7 | (2, 7] | open at 2, closed at 7 |
Parentheses = open (endpoint excluded). Square brackets = closed (endpoint included). ∞ always gets a parenthesis — infinity is never “reached.”
Compound Inequalities
AND (intersection): both conditions must hold simultaneously.
x > 2 AND x < 7 → 2 < x < 7 → (2, 7)
OR (union): at least one condition holds.
x < −1 OR x > 3 → (−∞, −1) ∪ (3, ∞)
Absolute Value
The absolute value of a number is its distance from zero on the number line — always non-negative.
|5| = 5
|−5| = 5
|0| = 0
Formally:
|x| = x if x ≥ 0
|x| = −x if x < 0
Properties
|ab| = |a||b|
|a/b| = |a|/|b|
|a + b| ≤ |a| + |b| (triangle inequality)
The triangle inequality is one of the most useful facts in mathematics — it says the distance between two points can never exceed the sum of the distances via a third point.
Solving Absolute Value Equations
|x| = k means x = k or x = −k (two cases, as long as k ≥ 0).
|2x − 3| = 7
Case 1: 2x − 3 = 7 → x = 5
Case 2: 2x − 3 = −7 → x = −2
Always check both solutions in the original equation.
If k < 0: no solution (absolute value is never negative). If k = 0: one solution (x = whatever makes the inside zero).
Solving Absolute Value Inequalities
Two patterns — get them straight:
|x| < k (distance less than k) → x is within k of zero → AND condition:
|x| < k ↔ −k < x < k ↔ (−k, k)
|2x − 1| < 5
−5 < 2x − 1 < 5
−4 < 2x < 6
−2 < x < 3
|x| > k (distance greater than k) → x is outside k of zero → OR condition:
|x| > k ↔ x < −k OR x > k
|2x − 1| > 5
2x − 1 < −5 OR 2x − 1 > 5
x < −2 OR x > 3
The intuition: “less than” absolute value = between two bounds. “Greater than” = outside two bounds.
Number Line Representation
Every inequality or absolute value solution can be drawn on a number line:
- Open circle ○ at an endpoint = excluded (strict inequality < or >)
- Closed circle ● at an endpoint = included (≤ or ≥)
- Arrow or shaded region shows the solution set
Visualising inequalities on a number line makes compound inequalities and intersections/unions immediate.
Applications
Error tolerance: |measured − actual| ≤ 0.5 means the measurement is within half a unit of the true value. Absolute value naturally captures symmetric error bounds.
Optimization constraints: inequalities define feasible regions. In linear programming, you solve for the best value of an objective within a set of inequality constraints. The solution always lies at a corner of the feasible region.
Distance between points: |a − b| is the distance between a and b on the number line. This generalises to |a − b| as the distance between vectors in higher dimensions.