Inequalities

Inequality Symbols

An inequality compares two expressions that are not necessarily equal. There are four inequality symbols:

A handy memory tip: the open end of the symbol always points toward the larger value. Think of > as a hungry crocodile eating the bigger number.

Solving Linear Inequalities

Solve inequalities almost exactly like equations, with one critical rule:

When you multiply or divide both sides by a negative number, flip the inequality sign.

Example 1: 2x + 3 < 11

1. Subtract 3: 2x < 8
2. Divide by 2: x < 4

Example 2: 3x - 7 > 2

1. Add 7: 3x > 9
2. Divide by 3: x > 3

Example 3 (negative divisor): -4x ≥ 12

1. Divide by -4 and flip the sign: x ≤ -3

Why does the sign flip? Multiplying both sides by -1 reverses the number line. If 2 < 5, then -2 > -5.

Worked Examples with Two Steps

Example: 5x + 2 ≤ 3x + 10

1. Subtract 3x: 2x + 2 ≤ 10
2. Subtract 2: 2x ≤ 8
3. Divide by 2: x ≤ 4

Example: -2x + 1 > 7

1. Subtract 1: -2x > 6
2. Divide by -2, flip sign: x < -3

Representing on a Number Line

Number lines give a visual representation of solution sets:

• Open circle (○) — the endpoint is NOT included: used with > or <
• Closed (filled) circle (●) — the endpoint IS included: used with ≥ or ≤
• An arrow extends in the direction of the solution

Examples:

• x > 4: open circle at 4, arrow pointing right
• x ≤ -1: closed circle at -1, arrow pointing left
• 2 ≤ x < 6: closed circle at 2, open circle at 6, shaded region between them

●─────────○
2   3   4   5   6    (2 ≤ x < 6)

Compound Inequalities

A compound inequality combines two conditions using "and" (both must hold simultaneously).

Example: -3 ≤ x < 2

This means x is at least -3 and less than 2. Both conditions apply to the same x.

Values that satisfy this: -3, -2.5, -1, 0, 1, 1.99 (but NOT 2, and NOT -4)

Example: Find the integers that satisfy 1 < x ≤ 5

• x must be greater than 1 (not including 1) and at most 5 (including 5)
• Integers: 2, 3, 4, 5

Inequalities vs Equations

Integer Solutions

When asked for integer solutions (whole numbers), list all integers in the range.

Find all integers satisfying -2 < x ≤ 3: -1, 0, 1, 2, 3

Note: -2 is excluded (strict inequality), 3 is included (≤).

Common Mistakes

• Forgetting to flip the sign when dividing or multiplying by a negative — this is the most common error.
• Using the wrong circle type on a number line — open circle for strict (<, >), closed for or-equal-to (≤, ≥).
• Including the endpoint when it should be excluded — re-read the inequality symbol carefully.
• Treating inequalities exactly like equations and ignoring the flip rule.
• Solving compound inequalities incorrectly by applying operations to only one part.

Tips and Tricks

• After solving, test a value in your solution set: substitute it into the original inequality to check it works.
• For compound inequalities like -3 ≤ 2x + 1 < 7, apply operations to all three parts at once:
  • Subtract 1: -4 ≤ 2x < 6
  • Divide by 2: -2 ≤ x < 3
• Always write the smaller value on the left in a compound inequality: write -2 ≤ x < 5, not 5 > x ≥ -2.
• On a number line, arrows always extend outward — a solution like x < 4 has the arrow going left.
• If the question asks for the solution on a number line and you are unsure of the circle type, ask: "Does this x-value itself satisfy the inequality?" If yes, filled; if no, open.