Inequalities

1Summary 重點摘要

Compare two numbers using >, <, ≥, ≤. Solving inequality is similar to equation, but multiply/divide by negative flips sign.

2Vocabulary cards Vocabulary

3Step by step Steps

1. 1. What is an inequality?An equation states that two sides are equal; an inequality compares them, using four symbols: > greater than, < less than, ≥ greater than or equal, ≤ less than or equal. For example, x > 3 means any number above 3 — 3.1, 4 or 100 all qualify. A helpful memory aid: the wide, open side of the symbol faces the larger number. Unlike equations, the solution is usually a range of values, not a single number.
2. 2. Solving: add or subtractAdding or subtracting the same number on both sides never changes the direction of the inequality, exactly as with equations. For x + 2 > 5, subtract 2 from both sides to get x + 2 − 2 > 5 − 2, that is x > 3. The goal is the same as solving equations: isolate the unknown. A common error is flipping the sign during addition or subtraction — remember, only multiplying or dividing by a negative number reverses it, never adding or subtracting.
3. 3. Solving: multiply or divide by a positiveMultiplying or dividing both sides by a positive number also leaves the direction unchanged. For 2x > 6, divide both sides by 2 to get x > 3; for x4 ≥ 2, multiply both sides by 4 to get x ≥ 8. These are the safest operations because no flip is needed. To check, test a value inside the range: x = 4 in 2x > 6 gives 8 > 6, which holds, so the direction was kept correctly.
4. 4. Solving: multiply or divide by a negative (KEY)Multiplying or dividing both sides by a negative number forces the inequality to reverse — the key rule of this topic. For −2x > 6, dividing by −2 gives x < −3, not x > −3. The reason is that a negative factor mirrors the number line and reverses the order: 2 < 5, but after multiplying by −1 it becomes −2 > −5. Forgetting this flip is the most common exam mistake, so take extra care whenever a negative coefficient appears.
5. 5. Showing the solution on a number lineSolutions are displayed on a number line: for x > 3, draw an open circle at 3 (3 itself excluded) with an arrow to the right; for x ≥ 3, draw a filled dot (3 included). The arrow rule is 'greater goes right, less goes left'. Two frequent errors are filling the circle for a strict > and pointing the arrow the wrong way. To check, pick a number from the arrow's side and confirm it satisfies the original inequality.
6. 6. Checking and applyingProper checking uses two values: one inside the solution and one outside. Since x > 3 solves x + 2 > 5, test x = 4: 6 > 5 ✓; then x = 2: 4 > 5 ✗, confirming 2 lies outside the solution set. For word problems, define the unknown, translate the sentence into an inequality, and after solving respect real-life limits — a number of people must be a positive integer, so state the answer as a whole-number range and answer what the question asks.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• > greater than, < less than, ≥ ≥, ≤ ≤
• Add/subtract doesn't change direction
• Multiply/divide positive doesn't change direction
• Multiply/divide negative MUST flip!
• Number line: open circle not included, filled circle included