Inequalities — Mathematics Knowledge System

Overview

An inequality compares two expressions using <, >, ≤, or ≥ instead of an equals sign, and its solution is typically a whole RANGE of values rather than a single number. Solving a linear inequality uses the same balance-scale operations as solving an equation, with one crucial extra rule: multiplying or dividing both sides by a negative number reverses the inequality's direction.

Formal Definition

Definition

A linear inequality in one variable has one of the forms:

ax+b < c, \quad ax+b \le c, \quad ax+b>c, \quad ax+b\ge c

General forms

Properties

Addition/subtraction property

a<b \iff a+c<b+c

Multiplication by a positive

a<b, \ c>0 \Rightarrow ac<bc

Multiplication by a negative flips the inequality

a<b, \ c<0 \Rightarrow ac>bc

Worked Examples

Subtract 5 from both sides.
$-2x>6$
Divide by -2, flipping the inequality sign since we're dividing by a negative.
$x<-3$

Answer: x < -3

Practice Problems

Difficulty 3/10

Solve: 3(x - 2) ≤ 5x + 4.

Common Mistakes

Common Mistake

Forgetting to flip the inequality sign when multiplying or dividing both sides by a negative number.

Whenever you multiply or divide both sides of an inequality by a NEGATIVE number, the direction of the inequality must reverse (< becomes >, and vice versa).

Summary

An inequality compares expressions with <, >, ≤, or ≥, and its solution is usually a range.
Solve using the same operations as equations, EXCEPT flip the inequality when multiplying/dividing by a negative.
Solutions are often expressed and graphed as intervals on a number line.

References

WebsiteWikipedia — Inequality (mathematics)