LINEAR INEQUALITIES Notes, Quizzes & Revision
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Mathematics โ Algebra
Subtopic: LINEAR INEQUALITIES (Age ~14 โ Kenyan context)
1. What is a linear inequality?
A linear inequality is like a linear equation but uses an inequality sign instead of โ=โ. It shows that one expression is less than or greater than another. Examples of inequality signs:
- < meaning "less than"
- > meaning "greater than"
- ≤ meaning "less than or equal to"
- ≥ meaning "greater than or equal to"
Example: 2x + 1 < 7 (read: "two x plus one is less than seven").
2. Rules for solving linear inequalities
- You may add or subtract the same number from both sides โ the inequality direction does NOT change.
- You may multiply or divide both sides by a positive number โ the direction does NOT change.
- If you multiply or divide both sides by a negative number, the inequality sign MUST be reversed. Example: if you divide by โ3, < becomes > and ≤ becomes ≥.
- Check answers by substituting a value from the solution into the original inequality.
3. Solving examples (step-by-step)
Example 1: Solve 2x + 3 > 7.
2x + 3 > 7
2x > 7 โ 3 (subtract 3 from both sides)
2x > 4
x > 4/2 (divide both sides by 2)
x > 2
2x > 7 โ 3 (subtract 3 from both sides)
2x > 4
x > 4/2 (divide both sides by 2)
x > 2
Solution in words: All numbers greater than 2.
Example 2: Solve โ3x ≤ 9.
โ3x ≤ 9
x ≥ 9 รท (โ3) (divide both sides by โ3 and flip the sign)
x ≥ โ3
x ≥ 9 รท (โ3) (divide both sides by โ3 and flip the sign)
x ≥ โ3
Note the ≤ changed to ≥ because we divided by a negative number.
4. Number line pictures
We show solutions using open (not included) or closed (included) circles:
(a) x < 3
Meaning: all values left of 3, not including 3.
(b) x ≥ โ1
Meaning: โ1 and all numbers greater than โ1.
5. Compound inequalities
Compound inequalities join two inequalities. Example:
Solve 1 < 2x + 3 ≤ 7
Step 1: subtract 3 from all three parts:
1 โ 3 < 2x + 3 โ 3 ≤ 7 โ 3
โ2 < 2x ≤ 4
Step 2: divide all parts by 2 (positive, so sign unchanged):
โ1 < x ≤ 2
1 โ 3 < 2x + 3 โ 3 ≤ 7 โ 3
โ2 < 2x ≤ 4
Step 2: divide all parts by 2 (positive, so sign unchanged):
โ1 < x ≤ 2
Solution: x is greater than โ1 and at most 2. In interval notation: (โ1, 2].
6. A simple word problem (Kenyan context)
Wambui has some money. She wants to buy a book that costs KSh 120 and still have more than KSh 80 left. If she starts with x shillings, write and solve an inequality for x.
She must have: x โ 120 > 80 (money after buying book must be more than 80)
x > 80 + 120
x > 200
x > 80 + 120
x > 200
So she needs more than KSh 200 to meet her goal.
7. Quick checklist when solving inequalities
- Isolate the variable by adding/subtracting first.
- Divide or multiply last; if you use a negative number, flip the sign.
- Draw a number line to show the solution clearly.
- Check one or two values from your solution in the original inequality.
8. Practice questions
- Solve: 3x โ 5 ≤ 10
- Solve: โ2x > 6
- Solve and draw on a number line: โ3 ≤ 4 โ x < 5
- Word problem: A matatu fare is at most KSh 120. If you have KSh y, write an inequality that shows you can pay the fare and keep at least KSh 20.
Answers:
- 3x โ 5 ≤ 10 โ 3x ≤ 15 โ x ≤ 5
- โ2x > 6 โ x < โ3 (sign flips when dividing by โ2)
- โ3 ≤ 4 โ x < 5 โ subtract 4: โ7 ≤ โx < 1 โ multiply by โ1 and flip signs: 7 ≥ x > โ1 โ write as โ1 < x ≤ 7 (order properly: โ1 < x ≤ 7)
- y โ 120 ≥ 20 โ y ≥ 140 (you need at least KSh 140 to pay and keep KSh 20)