Download §4-1 LINEAR INEQUALITIES

§4-1
LINEAR INEQUALITIES
Definition
Linear inequalities in one variable are inequalities which can be written in one of the
following forms: ax + b > 0
ax + b < 0
ax + b ≥ 0
ax + b ≤ 0 where a and b are real numbers.
Properties
The Addition Properties of Inequalities
If a > b and c is a real number then a + c > b + c.
If a < b and c is a real number then a + c < b + c.
Properties
The Multiplication Properties of Inequalities
If a > b and c is a positive real number then ac > bc.
If a < b and c is a positive real number then ac < bc.
If a > b and c is a negative real number then ac < bc.
If a < b and c is a negative real number then ac > bc.
Example 1
Solve the inequality, 5x + 15 > 0.
5x + 15
5x + 15 - 15
5x
5x
5
x
Solution
-7
Example 2
-6
-5
-4
-3
-2
-1
> 0
> 0 - 15
> -15
15
> −
5
> -3
0
1
2
3
4
5
6
7
Solve the inequality, 4x + 5 ≤ 6.
4x + 5
4x + 5 - 6
4x
4x
4
≤ 6
≤ 6-5
≤ 1
1
≤
4
1
x ≤
4
Solution
-1
0 .25 .5 .75
1
Copyright©2007 by Lawrence Perez and Patrick Quigley
Example 3
Solve the inequality, -2x - 4 < 0.
Solution
-7
Example 4
-6
-5
< 0
< 0+4
< 4
4
>
−2
> -2
-3
0
-4
-2
-1
1
2
3
4
5
6
7
Solve the inequality, -16x + 7 ≤ 2x - 2.
2x - 2
2x - 2 - 7
2x - 9
2x - 9 - 2x
-9
−9
≥
− 18
1
x ≥
2
Solution
-3
Example 5
-2x - 4
-2x - 4 + 4
-2x
−2 x
−2
x
-16x + 7
-16x + 7 - 7
-16x
-16x - 2x
-18x
− 18 x
−18
≤
≤
≤
≤
≤
-1
0
-2
.5
1
2
3
Solve the inequality, (x − 3)2 ≥ x 2 + 3.
(x − 3)2
2
x − 6x + 9
2
2
x − 6x + 9 − x
-6x + 9
-6x + 9 - 9
-6x
−6x
−6
x
Solution
-7
-6
-5
-4
-3
-2
-1
2
x +3
2
x +3
2
2
x +3− x
3
3-9
-6
−6
≤
−6
≤ 1
≥
≥
≥
≥
≥
≥
0
1
2
3
4
5
6
7
Copyright©2007 by Lawrence Perez and Patrick Quigley
§4-1
PROBLEM SET
Solve each linear inequality and graph the solution on a number line.
1.
3x > 36
2.
x + 14 < 64
3.
x - 7 ≥ 43
4.
x
≤8
4
5.
3y - 2 < -14
6.
-2x > 14
7.
2r + 1 ≤ 4r - 5
8.
2x
≥6
3
9.
-2(x - 3) > 8
11. 2(m - 5) ≥ -9
12.
x
− 5 ≤ 40
9
14. 13x + 57 > 182
15. 14 ≤ 3x - 4
10. 1 - 2t < 7
13.
5x
< 10
3
x
+ 1 > 22
3
16. 5(b - 3) ≥ 7
17.
19.
20. 2x + 1 ≤ 5
-3k ≥ -2
2 + 9x
≤ 12
6
18. 12x - 3 < 8x - 9
21.
7 − 2s
<2
3
22. 6s + 8 > 8 - 5s
23.
25. 4 + 6(r + 2) > 9
26. 7 - 2x < 1
27.
3t
− 10 ≥ 6
2
28. 0.3c + 1.5 ≤ 0.8c
29. 0.1(v - 8) < 10
30.
3w 5 w
− >
2 6 3
31. 41.7x - 13.2 ≤ 91.8
32.
34. 3t - 7 < 8t + 5
35. 1.83 ≥ 7x - 4.19
37. 3s + 1 < 7s - 15
38.
40. 8 - 9a ≥ 9a - 8
2
5
t +8≥ t
3
4
24. 6s + 3 ≥ 4s - 1
33. 6(k + 10) > 5(k + 14)
1 2
5
≤ r+
3 3
6
36.
r+
3y
+ 7 > 10
4
39.
2 - 2(7 - 2x) ≤ 3(3 - x)
41.
3
1
(2t − 4) > t
5
5
42.
4(1 - b) < 2 (b + 14)
a
−4
5
45.
5− x <
48.
2x
1
−2≥
3
5
43.
5x
+ 2 ≥ 3x − 1
2
44.
3a + 2 ≤
46.
x 2 1
− >
2 3 4
47.
2x 1 1
− ≤
3 2 4
49. 3 + 2(x + 5) > x + 5(x + 1) + 1
2x
−6
3
50. 3 - 5t < 18
Copyright©2007 by Lawrence Perez and Patrick Quigley
§4-1
PROBLEM SOLUTIONS
1.
x > 126
2.
x < 50
3.
x ≥ 50
4.
x ≤ 32
5.
y < -4
6.
x < -7
7.
r≥3
8.
x≥9
9.
x < -1
10.
t > -3
11.
m≥
13. x < 6
14.
x>
17. x > 63
18.
x<−
1
2
21.
x>
25.
r>−
7
6
125
13
3
2
1
2
15. x ≥ 6
12. x ≤ 405
16.
b≥
22
5
19.
k≤
2
3
20. x ≤ 2
22. s > 0
23.
x≤
70
9
24. s ≥ -2
26. x > 3
27.
t≥
32
3
28. c ≥ 3
5
7
31.
x≤
350
139
32.
t≤
96
7
12
5
35.
x≤−
43
50
36.
r≤
3
2
9
8
29. v < 9
30.
w≥
33. k > 10
34.
t >−
37. s > 4
38. y > 4
39. x ≤ 3
40.
a≤
42. b > -4
43. x ≤ 6
44.
a≤−
9
8
48.
x≥
12
5
41.
t>
45.
x>
33
5
46.
49.
x<
7
4
50. t > -3
x>
11
6
47.
x≤
15
7
33
10
Copyright©2007 by Lawrence Perez and Patrick Quigley