Inequalities

Chapter 4: The Real Number Line

Learning objectives

Solve linear and quadratic inequalities
Use interval notation
Apply the triangle inequality

Equations ask "what value works?" Inequalities ask "what range of values works?" The answer is no longer a single number but a set, usually an interval. This shift from point-answer to set-answer is the leap that prepares you for limits in calculus, feasibility regions in optimisation, and confidence intervals in statistics. An inequality is a statement comparing expressions using $<, >, \leq, \geq$ ; solving one means finding every value of the unknown that makes it true.

Rules for Inequalities

If $a < b$ then $a + c < b + c$ (add same number to both sides).

If $a < b$ and $c > 0$ , then $ac < bc$ (multiply by positive).

If $a < b$ and $c < 0$ , then $ac > bc$ (reverse inequality when multiplying by negative).

Triangle Inequality

For all real numbers $a, b$ : $|a + b| \leq |a| + |b|$

For compound inequalities like $a < x < b$ , apply operations to all three parts simultaneously. The solution is often expressed as an interval: $(a, b) = {x : a < x < b}$ .

(Pick "x > -3" for Constraint 1 and "x ≤ 5" for Constraint 2 in the widget. The widget shows three rows: each constraint shaded individually and their intersection.)

Try it

Solve $3x - 7 > 2x + 1$ by inspection, no negative-multiplier flips needed.
Solve $-2x + 5 \leq 11$ and pay attention to where the sign flips.
In the widget, set Constraint 1 to "x ≥ -4" and Constraint 2 to "x ≤ 4." What is the intersection in interval notation?
Solve $|2x - 1| < 5$ by rewriting as a compound inequality.

Try it in code

A trap to watch for

The most damaging error: forgetting to flip the inequality when multiplying or dividing by a negative. Solve $-3x > 6$ . Many beginners write $x > -2$ , wrong. The correct step is "divide both sides by $-3$ AND flip": $x < -2$ . The flip happens because order is defined via positivity (§3.2), and multiplying by a negative reverses what counts as "positive on this side." The fix: every time you see a coefficient that is negative on the variable you are isolating, write a big "FLIP" in the margin before you divide.

What you now know

You can solve linear and absolute-value inequalities, express solution sets in interval notation, and use the triangle inequality to bound sums. The next chapter introduces quadratic equations, the first nonlinear equations you will solve, and the discriminant that classifies their roots.

Quick check

References

Lang, S. (1971). Basic Mathematics. Springer. Chapter 3, §4, inequalities and the triangle inequality.
Spivak, M. (2008). Calculus, 4th ed. Publish or Perish. Chapter 1: the triangle inequality as the first nontrivial application of the order axioms.
Hardy, G. H., Littlewood, J. E., and Pólya, G. (1934). Inequalities. Cambridge. The canonical reference; the first chapter motivates inequality manipulation rules from elementary principles.