Active Maths 2 (Strands 1–5): Ch 16 Solutions

Chapter 16 Exercise 16.1

1

Q. 1. (i) x ≤ 2, x ∈ Z

210–1

(ii) x > 1, x ∈ N

321

(iii) x ≥ −4, x ∈ R

–2–3–4–5

(iv) 1 > x ∴ x < 1, x ∈ R

210–1

(v) x < −6, x ∈ Z

–5–6–7–8

(vi) x < 5, x ∈ N

54321

(vii) −x ≥ 3, x ∈ R

x ≤ −3, x ∈ R

–2–3–4–5

(viii) −x < −5

∴ x > 5, x ∈ Z

7654

(ix) −5 > −x

∴ −x < −5

x > 5, x ∈ R

7654

(x) 0 < −x

−x > 0

∴ x < 0, x ∈ R

10–1–2

Q. 2. (i) x ≥ 4, x ∈ N

(ii) x ≤ −1, x ∈ Z

(iii) x ≥ 3, x ∈ R

(iv) x > 6, x ∈ R

(v) x ≤ 1, x ∈ Z

(vi) x ≤ 1, x ∈ N

Q. 3. Solve 4x > 8, x ∈ N

4x > 8 ÷4 both side

x > 2, x ∈ N

4321

Q. 4. Solve 10x − 4 ≥ 6, x ∈ R

+4 +4 10x − 4 ≥ 6

10x ≥ 10 ÷10 both side

x ≥ 1, x ∈ R

210–1

Q. 5. Solve 8 < 2 + x, x ∈ N

8 <2 + x

−2 −2 2 + x > 8

x > 6, x ∈ N

8765

Q. 6. Solve 3x + 8 ≤ x − 4, x ∈ Z

−x −x 3x + 8 ≤ x − 4

−8 −8 2x + 8 ≤ −4

2x ≤ −12 ÷2 both side

x ≤ −6, x ∈ Z

–5–6–7–8

Q. 7. Solve 2x + 3 > 6 + x, x ∈ Z

2x + 3 > 6 + x

x + 3 > 6

x > 3, x ∈ Z

5432

[swapping sides changes the order of the inequality sign]

2 Active Maths 2 (Strands 1–5): Ch 16 Solutions

Q. 8. Solve 2x − 5 ≥ 4x + 3, x ∈ R

2x − 5 ≥ 4x + 3

4x + 3 ≤ 2x − 5

2x + 3 ≤ −5

2x ≤ −8

x ≤ −4, x ∈ R

–3–4–5–6

Q. 9. Solve x + 4 < 2x − 2, x ∈ R

x + 4 < 2x − 2

2x − 2 > x + 4

x − 2 > 4

x > 6, x ∈ R

8765

Q. 10. Solve 4x − 3 ≤ 6x − 5, x ∈ N

4x − 3 ≤ 6x − 5

6x − 5 ≥ 4x − 3

2x − 5 ≥ −3

2x ≥ 2

x ≥ 1, x ∈ N

4321

Q. 11. Solve 2x − 5 < 3x, x ∈ Z

2x − 5 < 3x

3x > 2x − 5

x > −5, x ∈ Z

–3–4–5–6

Q. 12. Solve 8x − 5 > 9x − 5, x ∈ Z

8x − 5 > 9x − 5

9x − 5 < 8x − 5

x − 5 < −5

x < 0, x ∈ Z

10–1–2

Q. 13. Solve 3x − 6 < 10x + 1, x ∈ R 3x − 6 < 10x + 1 10x + 1 > 3x − 6 7x + 1 > −6 7x > −7 x > −1, x ∈ R

10–1–2

Q. 14. Solve 3(2x − 1) ≥ 5(3x + 5) − 1, x ∈ R 3(2x − 1) ≥ 5(3x + 5) − 1 6x − 3 ≥ 15x + 25 − 1 6x − 3 ≥ 15x + 24 15x + 24 ≤ 6x − 3 9x + 24 ≤ −3 9x ≤ −27 x ≤ −3, x ∈ R

–2–3–4–5

Q. 15. Solve 4(x − 2) > 3(2x − 5), x ∈ R 4(x − 2) > 3(2x − 5) 4x − 8 > 6x − 15 6x − 15 < 4x − 8 2x − 15 < −8 2x < 7 x < 3 1 __ 2 , x ∈ R

43

3½

21

Q. 16. Solve 3(7x − 4) − 2(x + 4) + x > 0, x ∈ N.

21x − 12 − 2x − 8 + x > 0 20x − 20 > 0 20x > 20 x > 1, x ∈ N

4321

Q. 17. Solve 3(2x − 10) ≤ 6x − 5(2x + 2), x ∈ Z

3(2x − 10) ≤ 6x − 5(2x + 2) 6x − 30 ≤ 6x − 10x − 10 6x − 30 ≤ −4x − 10 10x − 30 ≤ −10 10x ≤ 20 x ≤ 2, x ∈ Z 3210

3Active Maths 2 (Strands 1–5): Ch 16 Solutions

Q. 18. Solve x − 2(6x − 1) < 4(x + 1) − 2, x ∈ R

x − 2(6x − 1) < 4(x + 1) − 2

x − 12x + 2 < 4x + 4 − 2

−11x + 2 < 4x + 2

−15x + 2 < 2

−15x < 0

x > 0, x ∈ R

210–1

Q. 19. (i) p = number of people attending the meal

budget = €3,750

fixed cost = €400

cost per person = €35

∴ 35p + 400 ≤ 3,750

(ii) Solving 35p + 400 ≤ 3,750

35p + 400 ≤ 3,750

35p ≤ 3,350

p ≤ 95 5 __ 7

The maximum number of guests possible is 95.

Q. 20. (i) 75x money withdrawn after x weeks

(ii) After 4 weeks

1,000 − 75 × 4 = 1,000 − 300 = 700

€700 left

(iii) After x weeks

1,000 − 75x left

(iv) 1,000 − 75x < 400

(v) Solve (iv)

1,000 − 75x < 400

−75x < −600

x > 8

Siobhán will be on tour for 8 weeks

Q. 21. Option B is cheaper, i.e. less than Option A

Let x be the number of days.

∴ 47x + 76 < 45x + 100

2x < 24

x < 12

Option B is cheaper than A for the first 11 days (note at 12 days they are the same price and after that Option A is cheaper).

Q. 22. (i) 4.99x + 50 ≤ 200

(ii) Solve (i)

4.99x + 50 ≤ 200

4.99x ≤ 150

x ≤ 30.06

∴ Evan can buy 30 books

Q. 23. Company B is cheaper (i.e. less than company A).

(i) 38x + 65 < 31x + 115

7x + 65 < 115

7x < 50

x < 7 1 __ 7

∴ It is cheaper to rent from Company B for the first 7 days.

4 Active Maths 2 (Strands 1–5): Ch 16 Solutions

(ii) Day Company A (€) Company B (€)1 115 + 31 = 146 65 + 38 = 103

2 146 + 31 = 177 103 + 38 = 141

3 208 179

4 239 217

5 270 255

6 301 293

7 332 331

8 363 369

9 394 407

10 425 445

(iii)

420 6 8 10 12

350

400

450

500

300

250

200

150

100

50

0

Ch

arg

e (

€)

Days

Company A

Company B

(iv) 7 days as in (i)

Q. 24. Statement Always true Never true Sometimes trueIf a > b and b > c, then a > c If −a < 4 and b < −4, then a < b If a > b, then −a > −b If a > b and b < c, then a < c If 3a + 1 > 2, then a > 0 If 2b − 4 < 3b − 8, then b > 4 If a and b are both positive and

a < b, then 1 __ a < 1 __ b

Suggested answers for Q24 – student answers may vary

If a > b, b > c then a > c. Using a numberline

abb > c a > b

c

∴ a is always greater than c

If −a < 4 and b < −4 then a < b.

−a < 4 ⇒ a > −4

then a > −4 and b < −4

–3–4

a

b

–5

From the numberline a < b is never true.

If a > b then −a > −b. Multiplying by a negative changes

the direction of the inequality

5Active Maths 2 (Strands 1–5): Ch 16 Solutions

∴ if a > b then

(−1) × a < (−1) × b

i.e. −a < −b

∴ −a > −b is never true.

If a > b and b < c then a < c. Using a number line

a cc

* *

a > b

b

Two possible locations for c show that a is sometimes less than c.

If 3a + 1 > 2 then a > 0. Solving 3a + 1 > 2

3a > 1

a > 1 __ 3

∴ a > 0 always true

If 2b − 4 < 3b − 8 then b > 4.

2b < 3b − 4

−b < −4

b > 4

∴ b > 4 always true

If a, b both positive and a < b,

then 1 __ a < 1 __ b .

Let a = 3 and b = 4

then 1 __ 3 � 1 __ 4

also if a = 1 __ 2 and b = 3 __ 4

then 1 __ 1 _ 2 � 1 __

3 _ 4

i.e. 2 � 1 1 __ 3 thus 1 __ a < 1 __ b never

true.

Exercise 16.2Q. 1. (i) −3 ≤ x ≤ 2, x ∈ Z

(ii) 1 ≤ x ≤ 3, x ∈ N

(iii) 0 ≤ x ≤ 1, x ∈ Z

(iv) 1 ≤ x ≤ 5, x ∈ R

(v) −8 < x < −4, x ∈ R

(vi) 0 < x ≤ 4, x ∈ R

(vii) 8 ≤ x < 11, x ∈ R

Q. 2. 4 < x < 8, x ∈ N

87654

Q. 3. −3 ≤ x ≤ 1, x ∈ Z

10–1–2–3

Q. 4. −2 ≤ 2x < 2, x ∈ R

−2 ___ 2 ≤ 2x ___ 2 ≤ 2 __ 2

−1 ≤ x ≤ 1, x ∈ R

1–1 0

Q. 5. 5 ≤ 5x < 10, x ∈ R

5 __ 5 ≤ 5x ___ 5 < 10 ___ 5

1 ≤ x < 2, x ∈ R

Q. 6. 2 < x + 1 ≤ 5, x ∈ N

−1 −1 −1 2 < x + 1 ≤ 5

1 < x ≤ 4, x ∈ N

4321

Q. 7. −4 < 2x − 2 < 0, x ∈ Z

+2 +2 +2 −4 < 2x − 2 < 0

−2 < 2x < 2

− 2 __ 2 < 2x ___ 2 < 2 __ 2

−1 < x < 1, x ∈ Z

10–1

Q. 8. −7 ≤ 3x − 1 ≤ 2, x ∈ R

+1 +1 +1

−7 ≤ 3x − 1 ≤ 2

−6 ≤ 3x ≤ 3 ÷3

−2 ≤ x ≤ 1, x ∈ R

10–1–2

20 1

6 Active Maths 2 (Strands 1–5): Ch 16 Solutions

Q. 9. 9 ≤ 5x + 4 < 19, x ∈ R

−4 −4 −4 9 ≤ 5x + 4 < 19

5 ≤ 5x < 15 ÷5

1 ≤ x < 3, x ∈ R

321

Q. 10. −2 < −3 + x < 1, x ∈ R

+3 +3 +3 −2 < −3 + x < 1

1 < x < 4, x ∈ R

3 421

Q. 11. 10 < 8x + 2 ≤ 18, x ∈ N

−2 −2 −2 10 < 8x + 2 ≤ 18

8 < 8x ≤ 16 ÷8

1 < x ≤ 2, x ∈ N

321

Q. 12. 3 > −x > 1, x ∈ R

Multiplying by −1 will reverse the direction of the inequality signs

∴ −3 < x < −1, x ∈ R

123

Q. 13. 2 ≥ −x ≥ −1, x ∈ Z

2 ≥ −x ≥ −1 × −1

−2 ≤ x ≤ 1, x ∈ Z

0 1–1–2

Q. 14. 10 ≥ −2x + 4 ≥ 2, x ∈ R

−4 −4 −4 10 ≥ −2x + 4 ≥ 2

6 ≥ −2x ≥ −2 × −1

−6 ≤ 2x ≤ 2 ÷2

−3 ≤ x ≤ 1, x ∈ R

0 1–1–2–3

Q. 15. 20 ≥ 6 − 7x > 6, x ∈ R

−6 −6 −6

20 ≥ 6 − 7x > 6

14 ≥ −7x > 0 × −1

−14 ≤ 7x < 0 ÷7

−2 ≤ x < 0, x ∈ R

–1 0–2

Q. 16. −12 > −2 −5x ≥ −27, x ∈ N

+2 +2 +2

−12 > −2 −5x ≥ −27

−10 > −5x ≥ −25 ÷−5

2 < x ≤ 5, x ∈ N

5432

Q. 17. (i) A = {x | 3x − 6 < 9, x ∈ Z}

+6 +6 3x − 6 < 9

3x < 15 ÷3

x < 5, x ∈ Z

∴ A = {…, 2, 3, 4}

(ii) B = {a|5x + 3 ≥ −2, x ∈ Z}

−3 −3 5x + 3 ≥ −2

5x ≥ −5 ÷5

x ≥ −1, x ∈ Z

∴ B = {−1, 0, 1, 2, …}

(iii) A ∩ B = {−1, 0, 1, 2, 3, 4}

(iv) 43210–1

Q. 18. (i) C = {x | 2x − 8 ≤ 6, x ∈ N}

+8 +8 2x − 8 ≤ 6

2x ≤ 14 ÷2

x ≤ 7

∴ C = {1, 2, 3, 4, 5, 6, 7}

7Active Maths 2 (Strands 1–5): Ch 16 Solutions

(ii) D = {x | 3x − 2 ≥ 4, x ∈ N}

+2 +2 3x − 2 ≥ 4

3x ≥ 6 ÷3

x ≥ 2

∴ D = {2, 3, 4, …}

(iii) C ∩ D = {2, 3, 4, 5, 6, 7}

(iv) 765432

Q. 19. (i) E = {x | x + 3 > −4, x ∈ R}

−3 −3 x + 3 > −4

x > −7

–6 –5–7

(ii) F = {x | 2x + 1 > 3x + 2, x ∈ R}

−1 −1 2x + 1 > 3x + 2

−3x −3x 2x > 3x + 1

−x > 1

x < −1

–1 0–2

(iii) E ∩ F

–4 –3 –2 –1–5–6–7

Q. 20. (i) G = {x | 3(2x + 1) > − (x − 3), x ∈ R}

3(2x + 1) > − (x − 3)

−3 −3 6x + 3 > −x + 3

6x > −x +x 7x > 0 ÷7

x > 0

0 1

(ii) H = {x | 3(x + 2) ≥ 4 (x − 1), x ∈ R}

3(x + 2) ≥ 4(x − 1)

−6 −6 3x + 6 ≥ 4x − 4

3x ≥ 4x − 10 −4x −x ≥ −10 × −1

x ≤ 10

9 10

(iii) G ∩ H

100

Q. 21. (i) −2 < x − 3, x ∈ R

−2 < x − 3

+3 +3 x − 3 > −2

x > 1

(ii) 4x − 9 ≤ 7, x ∈ R

+9 +9 4x − 9 ≤ 7

4x ≤ 16 ÷4

x ≤ 4

Integer values of x which satisfy (i) x > 1 and (ii) x ≤ 4 are {2, 3, 4}

Q. 22. (i) 20 ≤ t + 9 ≤ 27

(ii) −9 −9 −9 20 ≤ t + 9 ≤ 27

11 ≤ t ≤ 18

The temperature of the water before heating could have ranged from 11°C to 18°C.

Q. 23. (i) Foothill zone

0 < x ≤ 1,000

(ii) Subalpine zone

1,700 < x ≤ 2,500

(iii) Snow zone

3,200 < x ≤ 4,100

8 Active Maths 2 (Strands 1–5): Ch 16 Solutions

Q. 24. (i) 70 ≤ x ≤ 120 dimpled ball

(ii) 70 ≤ x ≤ 120 ÷2

35 ≤ x ≤ 60 smooth ball

(iii) Range of dimples y

300 ≤ y ≤ 450

(iv) 300 ≤ y < 400

Q. 25. V = 58 − 10t

−58 −58 −58 10 < 58 − 10t < 40

−48 < −10t < −18 ÷10

−4.8 < −t < −1.8 × −1

4.8 > t > 1.8

Between 1.8 and 4.8 seconds

Q. 26. (i) 12 < 2p + 5 < 49, x ∈ Z

(ii) −5 −5 −5 12 < 2p + 5 < 49

7 < 2p < 44 ÷2

3.5 < p < 22, x ∈ Z

p can range from 4 to 21

Q. 27. (i) 5,000 ≤ 220x + 500 ≤ 9,500

(ii) −500 −500 −500 5,000 ≤ 220x + 500 ≤ 9,500

4,500 ≤ 220x ≤ 9,000 ÷220

20 5 ___ 11 ≤ x ≤ 40 10 ___ 11

The least number of days is 21 and the most is 40.

Q. 28. (i) Smallest distance: 10 − 6 = 4 km

(ii) Greatest distance: 10 + 6 = 16 km

(iii) 4 ≤ x ≤ 16 (v)

Amy4 km 6 km 6 km Shokri

Shokri

Office

Q. 29. (i) 5 × 8 = 40 less sold

∴ If 24 sold, original total is 24 + 40 = 64

(ii) c = 64 − 8n

where c = number sold and

n = number of weeks

(iii) 15 ≤ 64 − 8n ≤ 40

(iv) 15 ≤ 64 − 8n ≤ 40

−49 ≤ −8n ≤ −24

49 ≥ 8n ≥ 24

6.125 ≥ n ≥ 3

Therefore she was content on the 3rd, 4th, 5th and 6th Saturdays after the first Saturday.

Q. 30. (a) (i) Solve −2 < 5x + 3 ≤ 18, x ∈ R

−3 −3 −3 −2 < 5x + 3 ≤ 18

−5 < 5x ≤ 15 ÷5

−1 < x ≤ 3, x ∈ R

(ii) 2 3 410–1

(b) 35 < x ≤ 50 (must spend €35).

45 < y ≤ 60 €50 + €10 voucher = €60

Q. 31. (i) 250 ≤ 8x − 30 ≤ 322

(ii) +30 +30 +30 250 ≤ 8x − 30 ≤ 322

280 ≤ 8x ≤ 352 ÷8

35 ≤ x ≤ 44

44 questions must be answered correctly to guarantee victory.

9Active Maths 2 (Strands 1–5): Ch 16 Solutions

Revision Exercises

Q. 1. (i) Solve 3x − 1 ≤ x + 3, x ∈ Z 3x − 1 ≤ x + 3 3x ≤ x + 4 2x ≤ 4 x ≤ 2, x ∈ Z

–1 0 1 2

(ii) Solve 4x + 1 ≤ 13, x ∈ R 4x + 1 ≤ 13 4x ≤ 12 x ≤ 3, x ∈ R

3 42

(iii) Solve 2x + 1 < 10, x ∈ N

2x + 1 < 10

2x < 9

∴ x < 4 1 __ 2 , x ∈ N

4 5321

(iv) Solve x + 3 ≤ 5x − 17, x ∈ R x + 3 ≤ 5x − 17 3 ≤ 4x − 17 20 ≤ 4x

5 ≤ x x ≥ 5, x ∈ R

5 64

(v) Solve 1 + x ≥ 3x − 1, x ∈ N

1 + x ≥ 3x − 1

3x − 1 ≤ 1 + x

3x ≤ 2 + x

2x ≤ 2

x ≤ 1, x ∈ N

21

(vi) Solve 1 − 5x > −29, x ∈ R

1 − 5x > −29

−5x > −30

x < 6, x ∈ R

(vii) Solve 4 < 1 − x, x ∈ R 4 < 1 − x 3 < −x × −1 −3 > x x < −3, x ∈ R

–3 –2–4

Q. 2. (i) Solve 5 ≤ 2x + 1 ≤ 11, x ∈ N

−1 −1 −1 5 ≤ 2x + 1 ≤ 11 4 ≤ 2x ≤ 10 ÷2 2 ≤ x ≤ 5, x ∈ N

4 532

(ii) Solve −3 < 2x − 1 ≤ 13, x ∈ Z

+1 +1 +1 −3 < 2x − 1 ≤ 13

−2 < 2x ≤ 14 ÷2

−1 < x ≤ 7, x ∈ Z

4 5 6 73210–1

(iii) Solve −1 < 3x + 2 ≤ 17, x ∈ R

−2 −2 −2 −1 < 3x + 2 ≤ 17 −3 < 3x ≤ 15 ÷3 −1 < x ≤ 5, x ∈ R

4 53210–1

(iv) Solve −9 ≤ 4x − 1 ≤ 11, x ∈ Z

+1 +1 +1

−9 ≤ 4x − 1 ≤ 11 −8 ≤ 4x ≤ 12 ÷4 −2 ≤ x ≤ 3, x ∈ Z

3210–1–2

(v) Solve 1 < 2x − 7 < 5, x ∈ N

+7 +7 +7 1 < 2x −7 < 5 8 < 2x < 12 ÷2

4 < x < 6, x ∈ N

6546 75

10 Active Maths 2 (Strands 1–5): Ch 16 Solutions

Q. 3. (i) A: 1 − 3x ≥ −8, x ∈ N

1 − 3x ≥ −8

−3x ≥ −9

x ≤ 3

A = {1, 2, 3}

(ii) B: −11 ≤ 3 + 7x < 31, x ∈ Z

−3 −3 −3 −11 ≤ 3 + 7x < 31

−14 ≤ 7x < 28

−2 ≤ x < 4

B = {−2, −1, 0, 1, 2, 3}

(iii) C: −14 < 5x + 1 < 26, x ∈ Z

−1 −1 −1 −14 < 5x + 1 < 26

−15 < 5x < 25 ÷5

−3 < x < 5

C − {−2, −1, 0, 1, 2, 3, 4}

(v) D: −8 ≤ 7x − 1 ≤ 20, x ∈ Z

+1 +1 +1 −8 ≤ 7x − 1 ≤ 20

−7 ≤ 7x ≤ 21

−1 ≤ x ≤ 3

D = {−1, 0, 1, 2, 3}

(vi) E: −5 < 1 − 2x ≤ 3, x ∈ Z

−1 −1 −1 −5 < 1 − 2x ≤ 3

−6 < −2x ≤ 2 ÷−2

3 > x ≥ −1

OR −1 ≤ x < 3

E = {−1, 0, 1, 2}

Q. 4. (i) Solve 2x + 4 < 12, x ∈ N

2x + 4 < 12

2x < 8

x < 4

The set of positive integers less than four.

(ii) −3x + 1 ≤ 4, x ∈ Z −1 −1 −3x + 1 ≤ 4 −3x ≤ 3 ÷−3 x ≥ −1 The set of integers greater than or

equal to negative one.

(iii) 1 < 2x + 3 < 13, x ∈ R −3 −3 −3 1 < 2x + 3 < 13 −2 < 2x < 10 ÷2 −1 < x < 5 The set of real numbers greater

than negative one but less than five. (iv) Solve −3 ≤ 3 − 2x < 2, x ∈ R −3 −3 −3 −3 ≤ 3 − 2x < 2 −6 ≤ −2x < −1 ÷2 −3 ≤ −x < − 1 __ 2 × −1

3 ≥ x > 1 __ 2

i.e. 1 __ 2 < x ≤ 3

The set of real numbers greater than a half but less than or equal to three.

(v) Solve −50 < 15x − 200 ≤ 520, x ∈ N

+200 +200 +200 −50 < 15x −200 ≤ 520 150 < 15x ≤ 720 ÷15 10 < x ≤ 48 The set of positive integers greater

than ten but less than or equal to forty-eight.

Q. 5. E: 2x − 1 ≤ 7, x ∈ N +1 +1 2x −1 ≤ 7 2x ≤ 8 x ≤ 4, x ∈ N E = {1, 2, 3, 4}

F: 5 − 4x ≤ 1, x ∈ N 5 − 4x ≤ 1 −4x ≤ −4 ÷−4 x ≥ 1 ∴ F = {1, 2, 3, 4, …} E ∩ F = {1, 2, 3, 4}

11Active Maths 2 (Strands 1–5): Ch 16 Solutions

Q. 6. (i) P: 1 − 6x < 16, x ∈ R

1 − 6x < 16

−6x < 15

−6x ____ −6 > 15 ___ −6

x > −2 1 __ 2 , x ∈ R

–2–3 2½

(ii) Q: 23 − 4x ≥ 3, x ∈ R

23 − 4x ≥ 3

−4x ≥ −20 ÷ –4

x ≤ 5, x ∈ R

5 64

(iii) P ∩ Q

5 6432

2½

Q. 7. If x ≤ y, where x, y ∈ R

(i) +5 +5 x − 5 ≤ y − 5

x ≤ y True

(ii) 5x ≤ 5y ÷5

x ≤ y true

(iii) x ___ −5 ≤

y ___ −5

Multiplying both sides by −5 gives

x ≥ y

∴ x ___ −5 ≤ y ___ −5 Not true

(iv) x __ 5 ≤ y __ 5 ×5

x ≤ y True

Q. 8. If x > y and x, y ∈ R.

(i) −3 −3 x + 3 > y + 3

x > y True

(ii) +3 +3 x − 3 > y − 3

x > y True

(iii) x __ 3 > y __ 3 ×3

x > y True

(iv) −3x > −3y ÷−3

x < y

∴ −3x > −3y Not true

If x > y, is x2 > y2 always?

No, e.g. if x = −1 and y = −2

then x2 = 1, y2 = 4

and 1 � 4

Q. 9. Why is it incorrect to write 5 < x < −4, x ∈ R?

Because this means that x is less than −4 but greater than 5, which is impossible.

Q. 10. Hamza gets more than €5 but less than €10.

∴ Possible amounts are:

€5.50, €6.00, €6.50, €7.00, €7.50, €8.00, €8.50, €9.00, €9.50

Q. 11. Séamus 178 cm

Nóirín 167 cm

(a)

Séamus

167 cm 178 cm

Nóirín

Susie

(b) Susie’s height has an infinite range of possibilities, as height is a continuous variable.

Q. 12. 2(x − 16) < x + 11

2x − 32 < x + 11

x < 43

Bill must be younger than 43

Q. 13. Perimeter: 2(2x − 1) + 2x

10 < 2(2x − 1) + 2x < 16

10 < 4x − 2 + 2x < 16

+2 +2 +2

10 < 6x −2 < 16

12 < 6x < 18 ÷6

2 < x < 3

Area: x(2x − 1)

12 Active Maths 2 (Strands 1–5): Ch 16 Solutions

if x = 2 Area = 2(2 × 2 − 1)

= 2 × 3

= 6

if x = 3 Area = 3(2 × 3 − 1)

= 3 × 5

= 15

∴ The area is greater than 6 but less than 15 square units

Q. 14. (i) 2x + 9 > 20; this is because the two CDs and the DVD cost more than the €20 Sheena had.

2x + 9 > 20

2x > 11

x > 5.50

(ii) 2x + 7 ≤ 20 this is because Sheena could afford both CDs and the perfume from the €20.

2x + 7 ≤ 20

2x ≤ 13

x ≤ 6.50

(iii) From (i) CD > €5.50.

From (ii) CD ≤ €6.50.

As the CD price is a whole number, a CD costs €6.00.

Q. 15. x + x + 3 > 12

2x + 3 > 12

2x > 9

x > 4.5

∴ x = 5 (given x is a whole number)

Q. 16. (i) 1 2 30–1–2–3–4

1 2 30–1–2–3–4

(ii) −4 ≤ x ≤ 1, x ∈ Z

−2 ≤ x ≤ 2, x ∈ Z

Students’ answers will vary.

Q. 17. Width = x, length = x + 4

Perimeter: 2(x + 4) + 2x = 2x + 8 + 2x

= 4x + 8

Assuming width >0

−8 −8 −8 8 < 4x + 8 < 30

0 < 4x < 22 ÷4

0 < x < 5.5

The width is greater than 0 but less than 5.5 cm.

Q. 18. Sid’s amount = x

Fionn’s amount = x + 12

x + x + 12 < 60

2x + 12 < 60

2x < 48

x < 24

Sid has less than €24.