Reference · module 3
Inequalities
Everything this module explains, on one page.
Greater and less
Read inequality signs and tell which numbers fit.
- x > 3 means x is greater than 3; x < 3 means less than 3. The wide open end of the sign faces the bigger side, like a mouth eating the larger number.
- A line under the sign adds or equal: x ≥ 3 includes 3 itself, while x > 3 does not. At most 25 is ≤ 25, and at least 18 is ≥ 18.
- An equation usually has one answer, but an inequality has a whole range. x > 3 is true for 4, 3.5, 100 and infinitely many other numbers.
The mistake you are about to make
at most 25 → x < 25
at most 25 → x ≤ 25
At most sounds like below, so the strict sign seems right. At most allows exactly 25, so the sign needs the line: ≤.
Solving inequalities
Solve linear inequalities like equations, with one extra rule.
- Most steps are the same as for equations: 2x + 3 < 11 becomes 2x < 8, then x < 4. Adding, subtracting, and dividing by a positive number keep the sign.
- The one new rule: multiplying or dividing by a negative number flips the sign. −2x < 6 becomes x > −3. Check with x = 0: −2 × 0 = 0 is less than 6.
- The answer is a range, so write it as an inequality with x alone: x ≥ 2, or on a number line with a filled dot for ≤ and ≥ and an empty one for < and >.
The mistake you are about to make
−2x < 6, x < −3
−2x < 6, x > −3
Dividing works exactly as in equations, so the sign seems to stay. Dividing by a negative reverses the order of numbers, so the sign must flip.
Inequalities in real life
Turn limits like at most and at least into inequalities and solve them.
- Budgets, speed limits and minimum ages are inequalities. A bag may weigh at most 20 kg: w ≤ 20. You need at least 60 points to pass: p ≥ 60.
- When the answer counts things, it must be a whole number. If 12n ≤ 100 for boxes of 12 eggs, n ≤ 8.33, so the most whole boxes is 8, not 8.33.
- Check the boundary: 8 boxes use 96 eggs, which fits, and 9 boxes need 108, which does not. Testing both neighbours confirms the answer.
The mistake you are about to make
12n ≤ 100, n = 9
12n ≤ 100, n ≤ 8
8.33 rounds to 8 here, but many people round up out of habit. Rounding up breaks the limit: 9 boxes need 108 eggs.