Linear inequalities

An inequality compares two amounts that may not be equal.

A double inequality such as −2 < x ≤ 3 means x is bigger than −2 and at most 3.

Example: write "x is at least 4" as an inequality, and list the integers (whole numbers) that satisfy −2 < x ≤ 3.

1. "At least 4" means 4 or more: x ≥ 4
2. −2 isn't included, because of the < sign; 3 is, because of the ≤ sign
3. −1, 0, 1, 2, 3

On a number line:

• an open circle (not filled in) marks an end that is not included: < or >
• a closed circle (filled in) marks an end that is included: ≤ or ≥
• a line joins the two ends, or an arrow shows it goes on forever.

Example: describe how to show −2 < x ≤ 3 on a number line.

1. −2 is not included: an open circle at −2
2. 3 is included: a closed circle at 3
3. A line joining the two circles

Example: a number line has a closed circle at −1, with an arrow pointing right. Write the inequality.

1. −1 is included, and every number to its right
2. x ≥ −1

Solve an inequality like an equation, doing the same to both sides. There is one extra rule: when you multiply or divide by a negative number, flip the inequality sign.

Example: solve 3x − 5 < 13.

1. Add 5: 3x < 18
2. Divide by 3: x < 6

Harder example: solve 7 − 2x ≥ 15.

1. Subtract 7: −2x ≥ 8
2. Divide by −2, and flip the sign: x ≤ −4

Check with a value: x = −5 gives 7 + 10 = 17, which is ≥ 15.

A double inequality has three parts. Do the same to all three.

1. Undo each operation on all three parts.
2. Flip both signs if you multiply or divide by a negative.
3. If asked, list the integers in the range, checking each end.

Example: solve −3 ≤ 2x + 1 < 9 and list the integer solutions.

1. Subtract 1 from all three parts: −4 ≤ 2x < 8
2. Divide all three by 2: −2 ≤ x < 4
3. −2 is included (≤) but 4 is not (<)
4. −2, −1, 0, 1, 2, 3