(Sec 3) Linear Inequalities Solving Techniques Solutions · 2020. 12. 23. · 1 | P a g e (Sec 3)...
Transcript of (Sec 3) Linear Inequalities Solving Techniques Solutions · 2020. 12. 23. · 1 | P a g e (Sec 3)...
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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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5 | P a g e
(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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6 | P a g e
(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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7 | P a g e
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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11 | P a g e
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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12 | P a g e
(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