Inequalities (Copy)

Practice Questions — 2.6 Inequalities

Question 1

Represent on a number line:
(a) x < 4
(b) x ≥ −2
(c) −3 ≤ x < 2

Question 2

Solve each inequality:
(a) 3x < 2x + 4
(b) 5x − 7 ≥ 3x + 1
(c) 7 − 2x ≤ 11

Question 3

Solve the compound inequalities:
(a) −3 ≤ 2x + 1 < 7
(b) 4 < 3x − 2 ≤ 10
(c) −5 < 2 − x ≤ 4

Question 4

Write down the inequalities shown by the number line:
(a) A closed circle at −2, open circle at 5, shaded in between.
(b) An open circle at 1, shaded to the right.
(c) A closed circle at 0, shaded to the left.

Question 5

Solve and represent on a number line:
(a) 2x + 3 < 7
(b) 5 − x ≥ 1
(c) 3x − 4 ≤ 8

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Question 6

Solve each inequality and give integer values of x:
(a) 2x + 1 ≤ 7
(b) −3 < 4x ≤ 9
(c) 5x − 2 > 18

Question 7

Graphically represent each pair of inequalities in two variables:
(a) x ≥ 0, y ≥ 0
(b) y > x, y ≤ 4
(c) y ≥ −x + 2, y < 3

Question 8

List the inequalities that describe the shaded region bounded by the triangle with vertices (0, 0), (4, 0), and (0, 3).

Question 9

Solve for x:
(a) 4x − 7 < 9
(b) 2 − 3x ≥ −7
(c) −2x + 5 ≤ 1

Question 10

Form and solve inequalities from word problems:
(a) A bus ticket costs Rs. 35. Ali has Rs. 250. Write and solve an inequality to find how many tickets he can buy.
(b) A number increased by 7 is less than 20. Find the range of values.
(c) A shopkeeper sells pens at Rs. 18 each. If he wants to earn at least Rs. 270, write and solve an inequality for the number of pens.

Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course

Question 11

Solve and state the integer solutions:
(a) 3x − 4 < 14
(b) 2x + 5 ≥ 11
(c) −5 ≤ x − 3 < 7

Question 12

Represent graphically:
(a) y ≤ 2x + 1
(b) y > −½x + 3
(c) x ≥ 1

Question 13

List inequalities for the region inside the rectangle with vertices (0, 0), (6, 0), (6, 4), (0, 4).

Question 14

Solve for x and give integer values:
(a) 5x − 3 ≤ 17
(b) 2 − x > −1
(c) 6x + 2 ≥ 20

Question 15

Form inequalities:
(a) The perimeter of a rectangle is at most 40 cm. If its length is 3x and breadth is 2x, form an inequality in x.
(b) The sum of three consecutive integers is greater than 45. Form and solve an inequality.
(c) Twice a number plus 5 is less than or equal to 29. Solve for the number.

Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course

Answer key and explanations — 2.6 Inequalities

1. Number line representations

(a) x < 4 → shade all numbers left of 4 with open circle at 4.
(b) x ≥ −2 → shade all numbers right of −2 with closed circle at −2.
(c) −3 ≤ x < 2 → closed circle at −3, open circle at 2, shade between.

2. Solve linear inequalities

(a) 3x < 2x + 4 → x < 4.
(b) 5x − 7 ≥ 3x + 1 → 2x ≥ 8 → x ≥ 4.
(c) 7 − 2x ≤ 11 → −2x ≤ 4 → x ≥ −2.
Explanation: Same as solving equations, but reverse inequality sign if dividing by a negative.

3. Compound inequalities

(a) −3 ≤ 2x + 1 < 7
→ subtract 1: −4 ≤ 2x < 6
→ divide by 2: −2 ≤ x < 3.
(b) 4 < 3x − 2 ≤ 10
→ add 2: 6 < 3x ≤ 12
→ divide 3: 2 < x ≤ 4.
(c) −5 < 2 − x ≤ 4
→ subtract 2: −7 < −x ≤ 2
→ multiply by −1 (flip signs): 7 > x ≥ −2 → −2 ≤ x < 7.

4. Interpret number line

(a) −2 ≤ x < 5
(b) x > 1
(c) x ≤ 0

5. Solve and represent

(a) 2x + 3 < 7 → 2x < 4 → x < 2.
(b) 5 − x ≥ 1 → −x ≥ −4 → x ≤ 4.
(c) 3x − 4 ≤ 8 → 3x ≤ 12 → x ≤ 4.

Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course

6. Integer solutions

(a) 2x + 1 ≤ 7 → 2x ≤ 6 → x ≤ 3. (integers: …, 0, 1, 2, 3).
(b) −3 < 4x ≤ 9 → divide 4: −0.75 < x ≤ 2.25 → integers: 0, 1, 2.
(c) 5x − 2 > 18 → 5x > 20 → x > 4.

7. Graphical representation

(a) First quadrant region.
(b) Above line y = x, below/including line y = 4.
(c) Above line y = −x + 2, below line y = 3.

8. Triangle region (0,0), (4,0), (0,3)

Inequalities:
x ≥ 0, y ≥ 0, 3x + 4y ≤ 12 (equation of sloping line through (4,0) and (0,3)).

9. Solve

(a) 4x − 7 < 9 → 4x < 16 → x < 4.
(b) 2 − 3x ≥ −7 → −3x ≥ −9 → x ≤ 3.
(c) −2x + 5 ≤ 1 → −2x ≤ −4 → x ≥ 2.

10. Word problems

(a) 35n ≤ 250 → n ≤ 7.14 → max tickets = 7.
(b) x + 7 < 20 → x < 13.
(c) 18n ≥ 270 → n ≥ 15.

Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course

11. Integer solutions

(a) 3x − 4 < 14 → 3x < 18 → x < 6. Integers: all x ≤ 5.
(b) 2x + 5 ≥ 11 → 2x ≥ 6 → x ≥ 3. Integers: 3, 4, 5…
(c) −5 ≤ x − 3 < 7 → add 3: −2 ≤ x < 10. Integers: −2, −1, 0… 9.

12. Graphs

(a) y ≤ 2x + 1 → solid line slope 2, shade below.
(b) y > −½x + 3 → broken line slope −½, shade above.
(c) x ≥ 1 → vertical line at x = 1, shade right.

13. Rectangle region (0,0), (6,0), (6,4), (0,4)

Inequalities: 0 ≤ x ≤ 6, 0 ≤ y ≤ 4.

14. Solve

(a) 5x − 3 ≤ 17 → 5x ≤ 20 → x ≤ 4.
(b) 2 − x > −1 → −x > −3 → x < 3.
(c) 6x + 2 ≥ 20 → 6x ≥ 18 → x ≥ 3.

15. Word problem inequalities

(a) Perimeter: 2(3x + 2x) ≤ 40 → 10x ≤ 40 → x ≤ 4.
(b) Integers n, n+1, n+2 → sum = 3n + 3 > 45 → 3n > 42 → n > 14 → smallest integer ≥ 15.
(c) 2x + 5 ≤ 29 → 2x ≤ 24 → x ≤ 12.

Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course