Practice and Problem-Solving Exercises

A Practice

See Problem 1.

Write the inequality that represents the sentence.

10. The sum of a number and 5 is less than − 7 . negative 7 .
11. The product of a number and 8 is at least 25.
12. Six less than a number is greater than 54.
13. The quotient of a number and 12 is no more than 6.

See Problem 2.

Solve each inequality. Graph the solution.

14. − 12 ≥ 24 x negative 12 greater than or equal to 24 x
15. − 7 k < 63 negative 7 k less than 63
16. 8 a − 15 > 73 8 eh minus 15 greater than 73
17. 57 − 4 t ≥ 13 57 minus 4 t greater than or equal to 13
18. − 18 − 5 y ≥ 52 negative 18 minus 5 y greater than or equal to 52
19. 14 − 4 y ≥ 38 14 minus 4 y greater than or equal to 38
20. 4 ( x + 3 ) ≤ 44 4 open x plus 3 close less than or equal to 44
21. 2 ( m − 3 ) + 7 < 21 2 open m minus 3 close plus 7 less than 21
22. 4 ( n − 2 ) − 6 > 18 4 open n minus 2 close minus 6 greater than 18
23. − 2 ( w + 4 ) + 9 < − 11 negative 2 open w plus 4 close plus 9 less than negative 11

See Problem 3.

Solve each problem by writing an inequality.

24. The length of a picture frame is 3 in. greater than the width. The perimeter is less than 52 in. Describe the dimensions of the frame.
25. The lengths of the sides of a triangle are in the ratio 5 : 6 : 7. Describe the length of the longest side if the perimeter is less than 54 cm.
26. Find the lesser of two consecutive integers with a sum greater than 16.
27. The cost of a field trip is $220 plus $7 per student. If the school can spend at most $500, how many students can go on the field trip?

See Problem 4.

Is the inequality always, sometimes, or never true?

28. 9 ( x + 2 ) > 9 ( x − 3 ) 9 open x plus 2 close greater than 9 open x minus 3 close
29. 6 x − 13 < 6 ( x − 2 ) 6 x minus 13 less than 6 open x minus 2 close
30. − 6 ( 2 x − 10 ) + 12 x ≤ 180 negative 6 open 2 x minus 10 close plus 12 x less than or equal to 180
31. − 7 ( 3 x − 7 ) + 21 x ≥ 50 negative 7 open 3 x minus 7 close plus 21 x greater than or equal to 50
32. 3 + 5 x < 5 ( x + 1 ) 3 plus 5 x less than 5 open x plus 1 close
33. 2 ( x + 6 ) < 30 2 open x plus 6 close less than 30
34. 4 x − 8 > 1 + 4 ( x + 3 ) 4 x minus 8 greater than 1 plus 4 open x plus 3 close
35. 9 x + 2 ( 2 + x ) < 5 + 9 x 9 x plus 2 open 2 plus x close less than 5 plus 9 x

See Problems 5 and 6.

Solve each compound inequality. Graph the solution.

36. 2 x > − 10  and  9 x < 18 2 x greater than negative 10 , and , 9 x less than 18
37. 3 x ≥ − 12  and  8 x ≤ 16 3 x greater than or equal to negative 12 , and , 8 x less than or equal to 16
38. 6 x ≥ − 24  and  9 x < 54 6 x greater than or equal to negative 24 , and , 9 x less than 54
39. 7 x > − 35  and  5 x ≤ 30 7 x greater than negative 35 , and , 5 x less than or equal to 30
40. 4 x < 16  or  12 x > 144 4 x less than 16 , or , 12 x greater than 144
41. 3 x ≥ 3  or  9 x < 54 3 x greater than or equal to 3 , or , 9 x less than 54
42. 8 x > − 32  or  − 6 x ≥ 48 8 x greater than negative 32 , or , minus 6 x greater than or equal to 48
43. 9 x ≤ − 27  or  4 x ≥ 36 9 x less than or equal to negative 27 , or , 4 x greater than or equal to 36

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