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1 | P a g e

(Sec 3) Linear Inequalities – Solving Techniques Solutions

1(a) 5 4 3x

3 4 5x

3 9

3 ( )

x

x ans

‒3 ‒2 ‒1 0 1 2 3 4 5

(b) 8 2 16x

2 16 8x

2 8

4 ( )

x

x ans

0 1 2 3 4 5 6 7 8

(c) 11 2 1x

11 1 2x

10 2

5

5 ( )

x

x

x ans

0 1 2 3 4 5 6 7 8

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2(a) 6 2(5 7)x

6 10 14x

6 14 10x

20 10

2

2 ( )

x

x

x ans

(b) 2( 7) 1 3( 5)x x

2 14 1 3 15x x

14 1 15 3 2x x

30

30 ( )

x

x ans

(c) 1 3( 4) 2( 5)x x

1 3( 4) 2( 5)x x

1 3 12 2 10x x

1 12 10 2 3x x

1 5x

1 ( )

5x ans

3(a) 2 1

53

x

2 1 15x

2 15 1x

2 16

8 ( )

x

x ans

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(b) 1 1

1 ( 4)2 3

x x

1 11 2

2 3x x

1 11 2

3 2x x

2 31

6 6x x

51

6

6

5

6 ( )

5

x

x

x ans

(c) 1 1

( 5) ( 1)3 2

x x

1 5 1 1

3 3 2 2x x

5 1 1 1

3 2 2 3x x

10 3 3 2

6 6 6 6x x

13 1

6 6x

13

13 ( )

x

x ans

(d) 1 3 2

3 4 6

x x

4( 1) 9 2( 2)

12 12 12

x x

4( 1) 9 2( 2)x x

4 4 9 2 4x x

4 2 4 4 9x x

2 17 x

17 ( )

2x ans

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4(a) 16 2x and 2 3 17x

16 2x and 2 3 17x

8 x and 2 20x

8x and 10x

8 10x

4 5 6 7 8 9 10 11 12

(b) 5 4 6x x and 2 5 3 9x x

5 4 6x x and 2 5 3 9x x

4 x and 14 x

4x and 14x

0 2 4 6 8 10 12 14 16

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(c) 1 2( 4) 3( 5)x x and 1 ( 2) 2( 4)x x

1 2( 4) 3( 5)x x and 1 2 2 8x x

1 2 8 3 15x x and 1 2 8 2x x

1 8 15 3 2x x and 7 3x

8 5x and 7

3x

8

5x and

7

3x

31

5x and

12

3x

12 ( )

3x ans

0 1

3

2

3 1 1

1

3 1

2

3 2 2

1

3 2

2

3 3

5 (a) 6 4 8x

6 4, 4 8x x

2 , 4x x

2 4x

0 1 2 3 4 5 6 7 8

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(b) 5 1 3 7x

5 1 3 , 1 3 7x x

3 1 5, 3 1 7x x

3 6, 3 6x x

2, 2x x

2 2x

‒4 ‒3 ‒2 ‒1 0 1 2 3 4

(c) 1 3 2 10x

1 3 2, 3 2 10x x

1 2 3 , 3 10 2x x

3 12,

3 3x x

1 4x

0 1 2 3 4 5 6 7 8

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6(a)

1 2 3

2 3 4

x x

1 2 3

2 3 4

x x

1 2 30

2 3 4

x x

12, 6( 1) 4( 2) 3(3) 0x x

6 6 4 8 9 0x x

2 5 0x

2 5x

5 ( )

2x ans

(b)

4 151 3(3 5)

2

xx

4 15

1 9 152

xx

2( 9 14) 4 15x x

18 28 4 15x x

28 15 4 18x x

43 22x 43

22

211 ( )

22

x

x ans

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(c)

3 4 2 12

5 3

x x

3 4 2 12

5 3

x x

3 4 2 1

2 05 3

x x

15, 3(3 4) 5(2 1) 2 15 0x x

9 12 10 5 30 0x x

47 0x

47 x

47 ( )x ans

7(a)

1 3 2

2 4 3

x x

1 3 2

2 4 3

x x

1 3 2

02 4 3

x x

12, 6( 1) 3( 3) 4 2 0x x

6 6 3 9 8 0x x

3 7 0x

3 7x

7

3x

1

23

x

1 ( )x ans

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(b) 3 1 2 4 1

2 3 4

x x

3 1 2 4 1

2 3 4

x x

3 1 2 4 1

02 3 4

x x

12, 6(3 1) 4(2 4) 3 0x x

18 6 8 16 3 0x x

10 25 0x

10 25x

12

2x

1, 2 ( )x ans

8(a)

8 5 3 8x

8 5 3 8x

8 5 3, 5 3 8x x

8 3 5 , 5 8 3x x

11 5 , 5 5x x 11

, 15

x x

12 1 ( )

5x ans

(b) 1 5 3 4 12x x x

1 5 3, 5 3 4 12x x x x

1 3 5 , 5 4 12 3x x x x

2 6 , 9x x 1

, 93

x x

19 ( )

3x ans

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10 | P a g e

9(a) 8 47x and 27 3 1x

47

8x and 27 1 3x

75

8x and 26 3x

75

8x and

26

3x

75

8x and

28

3x

7 25 8 ( )

8 3x ans

(b) 1 4 1

4 5

x x and

3 1 1

4 7

x x

1 4 10

4 5

x x and

3 1 10

4 7

x x

20, 5( 1) 4(4 1) 0x x and

28, 7(3 1) 4( 1) 0x x

5 5 16 4 0x x and

21 7 4 4 0x x

11 9 0x and

17 11 0x

9 11x and

17 11x

9

11x and

11

17x

9 11

11 17x

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10(a) 4 15

1 5 32

xx x

4 151 5 3, 5 3

2

xx x x

1 3 5 , 10 6 4 15x x x x

2 6 , 10 4 15 6x x x

1, 6 9

3x x

1 3,

3 2x x

1 3

3 2x

(b)

62 2 7

3

xx x

6

2 2 7, 2 73

xx x x

2 7 2 , 6 21 6x x x x

9 , 6 6 21x x x

9 , 5 15x x

9 , 3x x

9 3x

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(c)

11 3 2 7

3 2 3

x x x

11 3 2 3 2 7,

3 2 2 3

x x x x

2( 11) 3(3 2 ), 3(3 2 ) 2( 7)x x x x

2 22 9 6 , 9 6 2 14x x x x

2 6 9 22, 9 14 2 6x x x x

8 31, 5 8x x

31 5,

8 8x x

5 31 ( )

8 8x ans