To learn inequalities quickly and correctly, think of them as number comparisons that describe a range of answers instead of just one answer. An equation such as x = 4 points to a value, but an inequality such as x > 4 describes every number greater than four. The aim is to read each symbol precisely, keep each operation logically valid and test a value in the original question. These eduKateSG Mathematics Quick Notes explain how to solve inequalities without confusing them with equations.
- Read the sign: > means greater than; < means less than; ≥ and ≤ include equality.
- Look for a range: most simple inequalities have many solutions, not a single number.
- Add or subtract: do the same thing to both sides without changing the comparison sign.
- Multiply or divide by a negative number: reverse the comparison sign.
- Show the result: use a number line or clear inequality statement.
- Test: substitute a value that should work and one that should not.
First, picture the comparison on a number line. x > 3 means values to the right of 3, such as 3.1 and 4, but not 3 itself. x ≥ 3 includes the boundary value 3. For a number-line diagram, use an open circle when the boundary is excluded and a filled circle when it is included. Always draw an arrow towards the permitted values. This picture helps you understand what a solution means before you start manipulating symbols.
Second, solve most inequalities like balanced equations while remembering that the statement is about order. Suppose x + 5 < 12. Subtract 5 from both sides to obtain x < 7. If x = 6, the original reads 11 < 12, which is true; if x = 8, it reads 13 < 12, which is false. Adding or subtracting the same number preserves the order of two quantities. Do one small substitution check to confirm the direction.
Third, understand the special rule for a negative multiplier. If 2 < 5, multiplying both sides by −1 produces −2 > −5. The smaller positive number becomes the larger negative number, so the inequality reverses. This matters when solving −2x > 6. Divide by −2 and reverse the sign: x < −3. Test x = −4: −2(−4) = 8 and 8 > 6 is true. Writing that check makes the rule easier to remember than repeating “flip the sign” without a reason.
Worked example: brackets and a negative variable
Solve 3 − 2x ≥ 11. Subtract 3 from both sides: −2x ≥ 8. Divide both sides by −2 and reverse ≥ to ≤: x ≤ −4. Check x = −4: 3 − 2(−4) = 11, so the boundary is included. Check x = −5: 3 − 2(−5) = 13, which is also at least 11. Check x = −3: 9 ≥ 11 is false. These three substitutions verify the boundary and the correct side of it.
Read the four signs correctly
- x > 2: x is greater than two; two is excluded.
- x < 2: x is less than two; two is excluded.
- x ≥ 2: x is greater than or equal to two; two is included.
- x ≤ 2: x is less than or equal to two; two is included.
When to reverse the sign
Reverse the inequality only when multiplying or dividing both sides by a negative number in the usual solving steps. Do not reverse it when adding a negative number or subtracting a positive one. For instance, from x − 3 > 4, adding 3 gives x > 7. There is no negative multiplication. If you rewrite the entire statement with sides exchanged, the symbol must also face the new direction: x > 7 is equivalent to 7 < x.
Mistakes that make an easy inequality go wrong
- Ending with one answer: x > 3 is an entire set of possible values.
- Forgetting equality: ≥ and ≤ include the boundary; > and < do not.
- Reversing after subtraction: changing the comparison is not necessary just because you subtract.
- Missing the negative divide: −3x ≤ 9 becomes x ≥ −3.
- Checking a rearranged line only: put test values into the original problem.
Mini-test: practise without looking back
- Solve x + 7 > 10.
- Solve −3x ≤ 12.
- Solve 5 − x < 2.
Answers and quick checks
1. x > 3; subtract seven from both sides. 2. x ≥ −4; dividing by −3 reverses the comparison. 3. x > 3; subtract five, then divide by −1 and reverse. For question three, x = 4 gives 5 − 4 = 1, which is less than 2.
A simple eight-minute study plan
Spend two minutes reading and drawing the four signs, three minutes solving one inequality with a positive coefficient and one with a negative coefficient, two minutes checking substitutions and one minute explaining aloud why multiplying by a negative reverses order. Repeat one new example tomorrow. Your goal is to recognise the right solution range, not to memorise a particular question.
Continue the Mathematics learning pathway
Keep the skill connected to your other Quick Notes: Linear Equations · Negative Numbers · Algebra · Coordinate Graphs. Choose one related topic and complete a question from memory before moving on.
For the longer Secondary 1 learning approach, read Secondary 1 Mathematics Tutor Clementi | Small Groups Tutorials. This reference page is unchanged.