Linear inequalities in one unknown
How to solve linear inequalities in one unknown, including compound inequalities and "greatest integer" questions. It is a short unit, but it is tested directly and leads to F5 linear programming.
What you will learn
- 14.1 Write word phrases as inequalities; show x > a, x ≥ a, x < a, x ≤ a on a number line.
- 14.2 Properties:
- 14.3 Solve, and show the solution on a number line.
- 14.4 Word problems.
You need first: Number and algebra basics
Key terms
| English | 中文 |
|---|---|
| inequality | 不等式 |
| greater than, less than | 大於，小於 |
| greater than or equal to | 大於或等於 |
| at least, at most | 最少，最多 |
| solution set | 解集 |
| number line | 數線 |
| integer | 整數 |
Standard steps
Solve a linear inequality
- Solve exactly like an equation: expand, move terms, collect.
- If you multiply or divide by a negative number, flip the sign.
- Draw the number line: hollow circle for > or <, solid dot for ≥ or ≤.
- For "greatest integer" or "least integer", read it off the number line.
Why the flip: 3 > 2, but −3 < −2. Multiplying by −1 swaps the order of numbers on the number line.
Safer habit: move the x-terms to the side where the coefficient stays positive, so you never divide by a negative.
Worked examples
Example 1: Solve the linear inequality 3 − 2x > 7.
−2x > 4
x < −2 divide by −2, flipExample 2: Solve (2x − 1)/3 ≤ x + 2, and find the least integer x.
2x − 1 ≤ 3(x + 2) multiply both sides by 3 (positive, no flip)
2x − 1 ≤ 3x + 6
−7 ≤ x move 2x to the right to keep x positive
x ≥ −7
least integer: −7Number line: solid dot at −7, arrow to the right.
Common mistakes
- Not flipping the sign when dividing by a negative.
- Flipping the sign when moving a term (moving never flips).
- Multiplying only one term by the denominator.
- Hollow and solid dots mixed up.
- "Greatest integer satisfying x < 4" answered as 4 (it is 3).
How the DSE tests it
- A(1): solve, then "how many integers satisfy…" or "find the greatest integer…". In the senior curriculum the same question uses "and" / "or" (compound inequalities).
- Paper 2: choose the number line that matches a solution.
- Word problems: "at least 300 tickets" (≥), "no more than $50" (≤).
Questions you might not ask in class
Why don't we flip when we move a term across?
Moving is really adding or subtracting the same number on both sides, and that keeps the order.
Why does 'at least' mean ≥ and not >?
"At least 18" includes 18 itself.
Is x ≥ −7 the same as −7 ≤ x?
Yes. They say the same thing read from different sides.