Grade 9 Inequalities: Short Notes

9 ශ්‍රේණිය · 21 පරිච්ඡේදය · අසමානතා

40%

of Grade 9 term tests include it (16 of 40)

marks in past papers (17 questions)

10 min

full lesson with quick checks

🎯 What you'll learn

Read and use the signs >, <, ≥ and ≤
Show an inequality on a number line with filled and empty circles
Solve linear inequalities, reversing the sign for a negative multiplier
Find integer solutions, and the largest or smallest one
Write and solve inequalities from word problems

Height limit at the park

A ride at a Colombo fun park has a sign: riders must be at least 120 cm tall. 120 cm is fine, 135 cm is fine, 119 cm is not. One rule, but endless allowed heights. Maths writes this with an inequality: h ≥ 120.

The four signs

• x > 3: x is greater than 3 (3 not allowed)• x < 3: x is less than 3• x ≥ 3: x is greater than or equal to 3 (3 allowed)• x ≤ 3: x is less than or equal to 3Tip: the sign's open mouth faces the bigger side. 'At least' means ≥ and 'at most' means ≤.

On a number line

A filled dot means the end number is included (≥, ≤). An empty circle means it is not (>, <). The thick line shows all the other solutions.

Solve it like an equation

You may add or subtract the same number on both sides, and multiply or divide both sides by the same positive number. The sign stays the same.x + 5 < 9 → x < 43x ≥ 12 → x ≥ 4The answer is a range of values, not one number.

✏️ Two steps

Solve 3x − 2 ≥ 10 and show the solution on a number line.

1Add 2 to both sides: 3x ≥ 12.
2Divide both sides by 3: x ≥ 4.
3Number line: filled dot at 4, line going right.

✅ x ≥ 4

Dividing by a negative flips the sign

Start with 2 < 6, which is true. Multiply both sides by −1: −2 and −6. But −2 > −6, so the sign must turn around.Rule: when you multiply or divide by a negative number, reverse the inequality sign.−3x < 12 → x > −4

✏️ A negative coefficient

Solve 5 − 2x ≥ 1 and list the positive integer solutions.

1Subtract 5: −2x ≥ −4.
2Divide by −2 and reverse the sign: x ≤ 2.
3Positive integers start at 1: x = 1, 2.

✅ x ≤ 2; {1, 2}

⚠️ Forgetting to flip

Writing −2x ≥ −4 → x ≥ 2 is a very common slip. Check with a number: x = 3 gives 5 − 6 = −1, which is not ≥ 1, so x ≥ 2 is wrong. Flip the sign whenever you divide or multiply by a negative. Adding or subtracting a negative does not flip it.

⚠️ End points and 'integers'

Examiners see wrong circles at the ends. Ask for each end: is this number allowed? Yes → filled dot; no → empty circle.Also read the question: for integer solutions mark only separate dots, not a thick line. Positive integers start at 1, not 0.

📌 Remember

• >, < leave the end number out; ≥, ≤ include it.• Number line: filled dot = included, empty circle = not included.• Solve like an equation, but reverse the sign when you multiply or divide by a negative.• Integer solutions: list or mark separate dots; positive integers start at 1.• 'At least' is ≥, 'at most' is ≤.

🧾 Quick revision sheet

x ≥ 3 in words	x is greater than or equal to 3.
Filled dot on a number line	The end number is included (≥ or ≤).
Empty circle on a number line	The end number is not included (> or <).
When does the sign reverse?	When you multiply or divide both sides by a negative number.
Solve −2x < 8	x > −4
'At most 5' as an inequality	x ≤ 5
Integers with −1 < x ≤ 2	0, 1, 2