Solve the inequality:
4x + 6 < 30
4x + 6 < 30
Worked Solution & Answer
Solve like an equation using inverse operations.
e.g. 3b - 2 > 10 ⇒ 3b > 12 ⇒ b > 4
x < 6
e.g. 3b - 2 > 10 ⇒ 3b > 12 ⇒ b > 4
x < 6
Answer: x < 6
Solving inequalities — Notes & Worked Examples Show / Hide
Explanation:
Solve linear inequalities using the exact same inverse operations as linear equations, keeping the inequality sign (<, >, ≤, ≥) in the middle instead of an equals sign. Never replace the inequality sign with an = sign in your final answer!
Crucial rule: if you multiply or divide both sides by a negative number, you must flip the inequality sign around (or avoid negative x terms altogether by adding the negative x term to the other side).
Example 1: Solve 5x − 7 > 2x + 8
Solution:
Subtract 2x from both sides: 3x − 7 > 8
Add 7 to both sides: 3x > 15
Divide both sides by 3: x > 5
Answer: x > 5
Example 2: Solve the double inequality −5 ≤ 3x + 4 < 19 and state the largest integer x can be.
Solution:
Subtract 4 from all three parts: −9 ≤ 3x < 15
Divide all three parts by 3: −3 ≤ x < 5
Since x < 5, the largest integer is 4.
Answer: −3 ≤ x < 5; largest integer = 4
Solve the inequality:
4x - 5 ≥ 11
4x - 5 ≥ 11
Worked Solution & Answer
Solve like an equation using inverse operations.
e.g. 3b - 2 > 10 ⇒ 3b > 12 ⇒ b > 4
x ≥ 4
e.g. 3b - 2 > 10 ⇒ 3b > 12 ⇒ b > 4
x ≥ 4
Answer: x ≥ 4
(a) Write down the inequality shown on the number line. (1) (b) Solve 4y – 13 ≤ y + 8. (2)
(a) Write down the inequality shown on the number line. [1 mark]
(b) Solve the inequality 4y - 13 ≤ y + 8 [2 marks]
Worked Solution & Answer
(a)
x > –3
Answer (a): x > –3
(b)
3y ≤ 21 M1
⇒ y ≤ 7 A1
⇒ y ≤ 7 A1
Answer (b): y ≤ 7
(a) Inequality shown on number line (2) (b) Solve 7a − 5 ≤ 3a + 28 (2)
(a) Write down the inequality shown on the number line. [2 marks]
(b) Solve the inequality 7a - 5 ≤ 3a + 28
Show clear algebraic working. [2 marks]
Show clear algebraic working. [2 marks]
Worked Solution & Answer
(a)
Solve like an equation using inverse operations. M1
e.g. 3m - 2 > 10 ⇒ 3m > 12 ⇒ m > 4
Answer: -2 < x ≤ 1 A1
e.g. 3m - 2 > 10 ⇒ 3m > 12 ⇒ m > 4
Answer: -2 < x ≤ 1 A1
Answer (a): -2 < x ≤ 1
(b)
4a ≤ 33 M1
⇒ a ≤ 8.25 A1
⇒ a ≤ 8.25 A1
Answer (b): a ≤ 8.25
(a) Solve the inequality 20 – 5x ⩽ 30. [3 marks]
15 (b) Represent x > –1 on the number line. [1 mark]
15 (b) Represent x > –1 on the number line. [1 mark]
(a) Solve the inequality 20 - 5x ≤ 30 [3 marks]
Answer
(b) Represent x > -1 on the number line. [1 mark]
Worked Solution & Answer
(a)
5x ⩽ 10 M2
⇒ x ⩾ –2 A1
⇒ x ⩾ –2 A1
Answer (a): Answer x ≥ -2
(b)
Open circle at –1 and line with arrow pointing right B1
Answer (b): Open circle at –1 with arrow pointing to the right
(a) Show the inequality –2 ⩽ x < 3 on the number line. (2) (b) Solve the inequality 4y + 7 < 16 (2)
(a) Show the inequality -2 ≤ x < 3 on the number line below. [2 marks]
(b) Solve the inequality 4y + 7 < 16 [2 marks]
Worked Solution & Answer
(a)
Filled circle at –2, open circle at 3, line joining them M1 A1
Answer (a): Filled circle at –2, open circle at 3, line joining them
(b)
4y < 16 – 7 M1
⇒ 4y < 9
⇒ y < 9/4 (2.25) A1
⇒ 4y < 9
⇒ y < 9/4 (2.25) A1
Answer (b): y < 2.25
(i) Solve the inequalities –7 ≤ 2x – 3 < 5 (3) (ii) On the number line, represent the solution set to part (i) (2)
(i) Solve the inequalities -7 ≤ 2x - 3 < 5 [3 marks]
(ii) On the number line, represent the solution set to part (i) [2 marks]
Worked Solution & Answer
(i)
–7 + 3 ≤ 2x < 5 + 3 M1
⇒ –4 ≤ 2x < 8 M1
⇒ –2 ≤ x < 4 A1
⇒ –4 ≤ 2x < 8 M1
⇒ –2 ≤ x < 4 A1
Answer (i): -2 ≤ x < 4
(ii)
–7 + 3 ≤ 2x < 5 + 3 M1
⇒ –4 ≤ 2x < 8
⇒ –2 ≤ x < 4 A1
⇒ –4 ≤ 2x < 8
⇒ –2 ≤ x < 4 A1
Answer (ii): Closed circle at –2, open circle at 4, line joining –2 to 4
(a) Solve the inequality.
4(x - 3) < x [3 marks]
4(x - 3) < x [3 marks]
(b) Show your answer to **part (a)** on the number line. [2 marks]
Worked Solution & Answer
(a)
Expand the bracket: 4x - 12 < x M2
Subtract x and add 12: 3x < 12 ⇒ x < 4. A1
Subtract x and add 12: 3x < 12 ⇒ x < 4. A1
Answer (a): x < 4
(b)
Draw an open (unshaded) circle at 4 (since the inequality is strict, <) with an arrow pointing to the left. M1 A1
Answer (b): Open circle at 4 with arrow pointing to the left.
(a) Inequality on number line (b) Integer values of y for −4 ≤ 2y < 6 (c) Solve 7t − 3 ≤ 2t + 31
(a) Write down the inequality shown on the number line. [1 mark]
(b) Write down all the possible values of y. [2 marks]
(c) Solve the inequality 7t - 3 ≤ 2t + 31
Show your working clearly. [2 marks]
Show your working clearly. [2 marks]
Worked Solution & Answer
(a)
x ≤ 2
Answer (a): x ≤ 2
(b)
−2 ≤ y < 3 M1
⇒ −2, −1, 0, 1, 2 A1
⇒ −2, −1, 0, 1, 2 A1
Answer (b): -2, -1, 0, 1, 2
(c)
5t ≤ 34 M1
⇒ t ≤ 6.8 A1
⇒ t ≤ 6.8 A1
Answer (c): t ≤ 6.8
(a) Factorise 25f − 10 (1) (b) Make y subject of c = 5y − h (2) (c) Solve 4x + 7 > 2 (2)
(a) Factorise 25f - 10 [1 mark]
(b) Make y the subject of the formula c = 5y - h [2 marks]
y =
(c) Solve the inequality 4x + 7 > 2 [2 marks]
Worked Solution & Answer
(a)
5(5f − 2)
Answer (a): 5(5f - 2)
(b)
c + h = 5y M1
⇒ y = (c + h) / 5 A1
⇒ y = (c + h) / 5 A1
Answer (b): y = y = c + h5
(c)
4x > −5 M1
⇒ x > −1.25 A1
⇒ x > −1.25 A1
Answer (c): x > -1.25