Grade 10 Algebraic Inequalities: Short Notes

10 ශ්‍රේණිය · 24 පරිච්ඡේදය · වීජීය අසමානතා

63%

of Grade 10 term tests include it (34 of 54)

132

marks in past papers (37 questions)

10 min

full lesson with quick checks

🎯 What you'll learn

Read and write inequalities with <, >, ≤ and ≥
Show an inequality on a number line and read one from a number line
Solve linear inequalities, reversing the sign for a negative multiplier
Find integer solutions: greatest, least or positive integers
Form and solve an inequality from a real situation

How far can Rs 500 take you?

A three-wheeler in Galle charges Rs 100 to start plus Rs 80 per km. You have Rs 500. You don't need the bill to be exactly Rs 500; it just must not go over. That's not an equation, it's an inequality: 100 + 80x ≤ 500.

Four signs

• x < 4: x is less than 4• x > 4: x is greater than 4• x ≤ 4: x is less than or equal to 4 (at most 4)• x ≥ 4: x is greater than or equal to 4 (at least 4)An inequality has many solutions, not just one. x < 4 is true for 3, 0, −10, 3.9...

On a number line

Top: x > −1. Open circle: −1 is not included. Bottom: x ≤ 3. Filled circle: 3 is included. The arrow shows every number on that side.

Solve it like an equation

You may do these to both sides and keep the sign:• add or subtract any number• multiply or divide by a positive number3x − 5 < 133x < 18 (add 5)x < 6 (divide by 3)

✏️ Greatest integer

Solve 4x + 3 ≤ 2x + 14 and find the greatest integer value of x.

1Subtract 2x from both sides: 2x + 3 ≤ 14
2Subtract 3: 2x ≤ 11
3Divide by 2: x ≤ 5.5
4The greatest integer that is at most 5.5 is 5.

✅ x ≤ 5.5; greatest integer 5

The one new rule

2 < 5 is true. Multiply both sides by −1: −2 and −5. Now −2 is bigger, so −2 > −5.When you multiply or divide by a negative number, reverse the sign: < becomes >, ≤ becomes ≥.−3x > 12 → x < −4

✏️ Dividing by a negative

Solve 7 − 2x ≥ 15 and show the solution on a number line.

1Subtract 7: −2x ≥ 8
2Divide by −2 and reverse the sign: x ≤ −4
3Check with x = −5: 7 + 10 = 17 ≥ 15 ✓
4Filled circle at −4, arrow to the left.

✅ x ≤ −4

⚠️ Forgetting to flip

In one O/L paper only about 3 in 10 candidates solved a linear inequality correctly. The usual slip: dividing by a negative and keeping the sign.Tip: check with one number. If you got x ≥ −4 for 7 − 2x ≥ 15, try x = 0: 7 ≥ 15 is false, so the sign is wrong.

Integer solutions

Exams often ask for integer values that satisfy an inequality.• Strict (< or >): the end number is not a solution.• Positive integers start at 1. Zero is neither positive nor negative.Example: 2x + 1 < 9 gives x < 4. Positive integer solutions: 1, 2, 3.

⚠️ Open or filled? 0 or 1?

Examiners report two small slips that cost marks:• a filled dot for < or > (it must be open), or an open dot for ≤ or ≥• listing 0 as a positive integer, or the end value of a strict inequality (2x + 1 < 6 gives x < 2.5, so 3 is not a solution).

📌 Remember

• <, > leave the end out (open circle); ≤, ≥ include it (filled circle).• Solve like an equation: add, subtract, multiply or divide both sides.• Multiply or divide by a negative: reverse the sign.• Positive integers start at 1; read 'greatest', 'least', 'at most', 'at least' carefully.• Check your answer with one number.

🧾 Quick revision sheet

x ≤ 7 in words	x is less than or equal to 7 (at most 7).
Open circle on a number line	The end value is not included: < or >.
When does the inequality sign reverse?	When you multiply or divide both sides by a negative number.
Is 0 a positive integer?	No. Positive integers are 1, 2, 3, ...
−4x < 20	x > −5
Greatest integer with x < 3	2 (3 itself is not allowed)

Now test yourself: 5 quick checks and 2 more worked examples

The full lesson checks your answers as you go, and Gedara Sir tracks which parts you've got.

📖 Start the 10-minute lesson free ▶ Practise 5 past-paper questions