Inequalities

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Quick recall

• Reverse the inequality when multiplying or dividing by a negative number.
• Apply the same operation to every part of a compound inequality.

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Formula Cards

1. Negative multiplication or division

$$-2x>6\quad\Rightarrow\quad x<-3$$

Reverse the inequality sign when multiplying or dividing by a negative number.

2. Compound inequalities

$$a<x<b$$

Solve both bounds; perform each operation on all three parts. Addition/subtraction and positive multiplication/division keep the direction unchanged.

Worked Examples

Example 1

Question: Solve −2x > 6.

Step 1: Divide both sides by −2 and reverse the sign.

$$x<\frac{6}{-2}$$

Step 2: Simplify.

$$x<-3$$

Answer: x < −3.

Example 2

Question: Solve 3x+5 ≤ 20.

Step 1: Subtract 5.

$$3x\le15$$

Step 2: Divide by positive 3, keeping the sign.

$$x\le5$$

Answer: x ≤ 5.

Example 3

Question: Solve 2 < 3x−1 < 11.

Step 1: Add 1 to all three parts.

$$3<3x<12$$

Step 2: Divide every part by 3.

$$1<x<4$$

Answer: 1 < x < 4.

Drill

Try each question before opening its answer.

Question 1

Solve $5-2x>1$.

• Show answer

  x<2

Question 2

Solve $4x+3\ge15$.

• Show answer

  x≥3

Question 3

Solve $-3x\le9$.

• Show answer

  x≥-3

Question 4

Solve $1\le2x-3\le7$.

• Show answer

  2≤x≤5

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