Practice and Problem-Solving Exercises

A Practice

See Problem 1.

Find all whole number solutions of each system using a table.

8. { y + 3 x ≤ 8 y − 3 > 2 x left brace . table with 2 rows and 1 column , row1 column 1 , y plus 3 x less than or equal to 8 , row2 column 1 , y minus 3 greater than 2 x , end table
9. { x + y < 8 3 x ≤ y + 6 left brace . table with 2 rows and 1 column , row1 column 1 , x plus y less than 8 , row2 column 1 , 3 x less than or equal to y plus 6 , end table
10. { y ≥ x + 2 3 y < − 6 x + 6 left brace . table with 2 rows and 1 column , row1 column 1 , y greater than or equal to x plus 2 , row2 column 1 , 3 y less than negative 6 x plus 6 , end table

See Problem 2.

Solve each system of inequalities by graphing.

11. { y ≤ 2 x + 2 y < − x + 1 left brace . table with 2 rows and 1 column , row1 column 1 , y less than or equal to 2 x plus 2 , row2 column 1 , y less than negative x plus 1 , end table
12. { y > − 2 x < 1 left brace . table with 2 rows and 1 column , row1 column 1 , y greater than negative 2 , row2 column 1 , x less than 1 , end table
13. { y ≤ 3 x ≤ 1 2 x + 1 left brace . table with 2 rows and 1 column , row1 column 1 , y less than or equal to 3 , row2 column 1 , x less than or equal to , 1 half , x plus 1 , end table
14. { y ≤ 3 x + 1 − 6 x + 2 y > 5 left brace . table with 2 rows and 1 column , row1 column 1 , y less than or equal to 3 x plus 1 , row2 column 1 , negative 6 x plus 2 y greater than 5 , end table
15. { x + 2 y ≤ 10 x + y ≤ 3 left brace . table with 2 rows and 1 column , row1 column 1 , x plus 2 y less than or equal to 10 , row2 column 1 , x plus y less than or equal to 3 , end table
16. { − x − y ≤ 2 y − 2 x > 1 left brace . table with 2 rows and 1 column , row1 column 1 , negative x minus y less than or equal to 2 , row2 column 1 , y minus 2 x greater than 1 , end table
17. { y > − 2 x 2 x − y ≥ 2 left brace . table with 2 rows and 1 column , row1 column 1 , y greater than negative 2 x , row2 column 1 , 2 x minus y greater than or equal to 2 , end table
18. { c ≥ d − 3 c < 1 2 d + 3 left brace . table with 2 rows and 1 column , row1 column 1 , c greater than or equal to d minus 3 , row2 column 1 , c less than , 1 half , d plus 3 , end table
19. { 2 x + y < 1 y > − 2 x + 3 left brace . table with 2 rows and 1 column , row1 column 1 , 2 x plus y less than 1 , row2 column 1 , y greater than negative 2 x plus 3 , end table

    See Problem 3.

20. You want to decorate a party hall with a total of at least 40 red and yellow balloons, with a minimum of 25 yellow balloons. Write and graph a system of inequalities to model the situation.

21. A gardener wants to plant at least 50 tulips and rose plants in a garden, but no more than 20 rose plants. Write and graph a system of inequalities to model the situation.

See Problem 4.

Solve each system of inequalities by graphing.

22. { y > 4 y < | x − 1 | left brace . table with 2 rows and 1 column , row1 column 1 , y greater than 4 , row2 column 1 , y less than vertical line x minus 1 vertical line , end table
23. { y < − 1 3 x + 1 y > | 2 x − 1 | left brace . table with 2 rows and 1 column , row1 column 1 , y less than negative , 1 third , x plus 1 , row2 column 1 , y greater than vertical line 2 x minus 1 vertical line , end table
24. { y > x − 2 y ≥ | x + 2 | left brace . table with 2 rows and 1 column , row1 column 1 , y greater than x minus 2 , row2 column 1 , y greater than or equal to vertical line x plus 2 vertical line , end table
25. { y ≤ − 4 3 x y ≥ − | x | left brace . table with 2 rows and 1 column , row1 column 1 , y less than or equal to negative , 4 thirds , x , row2 column 1 , y greater than or equal to negative vertical line x vertical line , end table
26. { 3 y < − x − 1 y ≤ | x + 1 | left brace . table with 2 rows and 1 column , row1 column 1 , 3 y less than negative x minus 1 , row2 column 1 , y less than or equal to vertical line x plus 1 vertical line , end table
27. { y > − 2 y ≤ − | x − 3 | left brace . table with 2 rows and 1 column , row1 column 1 , y greater than negative 2 , row2 column 1 , y less than or equal to negative vertical line x minus 3 vertical line , end table
28. { − 2 y < 4 x + 2 y > | 2 x + 1 | left brace . table with 2 rows and 1 column , row1 column 1 , negative 2 y less than 4 x plus 2 , row2 column 1 , y greater than vertical line 2 x plus 1 vertical line , end table
29. { − x ≥ 4 − y y ≥ | 3 x − 6 | left brace . table with 2 rows and 1 column , row1 column 1 , negative x greater than or equal to 4 minus y , row2 column 1 , y greater than or equal to vertical line 3 x minus 6 vertical line , end table
30. { y ≤ x − 4 y > | x − 6 | left brace . table with 2 rows and 1 column , row1 column 1 , y less than or equal to x minus 4 , row2 column 1 , y greater than vertical line x minus 6 vertical line , end table

B Apply

31. Think About a Plan The food pyramid suggests that you eat 4–6 servings of fruits and vegetables a day for a healthy diet. It also says that the number of servings of vegetables should be greater than the number of servings of fruits. Find the number of servings of fruits and vegetables that could make a healthy diet. Use whole numbers only.
    • How can you write two inequalities that model the information in the problem?
    • How can you use a graph to find combinations of fruits and vegetable servings that may help in having a healthy diet?
32. College Admissions An entrance exam has two sections, a verbal section and a mathematics section. You can score a maximum of 1600 points. For admission, the school of your choice requires a math score of at least 600. Write a system of inequalities to model scores that meet the school's requirements. Then solve the system by graphing.

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