Solving and Graphing Inequalities in Two Variables

Lewis has 50-cent stamps and 23-cent stamps. He needs to use stamps totalling at least $3.70 to mail a package. There is enough room for up to 10 stamps on the package. This situation can be represented by the following system of inequalities, where x = number of 50-cent stamps and y = number of 23-cent stamps. x + y ≤ 10 50x + 23y ≥ 370 Which statement is true?

• Lewis can use six 50-cent stamps and four 23-cent stamps.

• Lewis can use five 50-cent stamps and five 23-cent stamps.

• If Lewis uses six 50-cent stamps, he only needs to use two 23-cent stamp.

• If Lewis uses fourteen 23-cent stamps, he only needs to use one 50-cent stamp.

The members of a new band want to spend no more than $75 for at least 60 fliers to advertise their upcoming concert.  The cost color fliers is $1.50 each, the cost of black and white fliers is $0.75 each. Which of the following combinations is a reasonable solution?

• 26 color, 30 black and white

• 30 color, 35 black and white

• 60 color

• 30 color, 50 black and white

Solve each of the following systems of linear inequalities a graphing approach. Verify using a graphing calculator. Which of following are the vertices?

• (2, 0), (4, 5), (6, 0)

• (2, 0), (3, 5), (0, 6)

• (0, 2), (3.5, 4.5), (6, 2)

• (0, 0), (3, 5), (6, 0)

Ms. Loredo is making her special cantaloupe-and-honeydew salad for a family gathering she is hosting and does not want to spend more than $30 at the supermarket. Currently, cantaloupes cost $3 each and honeydew melons cost $2 each. Which inequality in slope-intercept form best represents the number of cantaloupes, x, and honeydew melon, y. (Write an inequality first, then solve for y.)

A student has started a lawn care business. He charges $15 per hour to mow lawns and $20 per hour for gardening. Because he is still in school, he is only allowed to work for at most 20 hours per week. His goal is to make at least $300 per week.

Which system of equations or inequalities model the situation?

• 15x + 20y > 20

  x + y        < 300

• 15x + 20y > 300

  x + y     < 20

A farmer has a field that can be used to grow two types of crops, corn and wheat. Each acre of corn requires 2 hours of labor and each acre of wheat requires 3 hours of labor. The farmer has a total of 40 hours of labor available. The farmer wants to grow at least 5 acres of corn and at least 8 acres of wheat. Write a system of linear inequalities to represent this situation.

• 2x + 3y >= 40, x <= 5, y <= 8.

• 2x + 3y <= 40, x <= 5, y <= 8.

• 2x + 3y >= 40, x >= 5, y <= 8.

• 2x + 3y <= 40, x >= 5, y >= 8

Write the system of inequalities:
Carlos works at a movie theater selling tickets. The theater has 300 seats and charges $7.50 for adults and $5.50 for children. The theater expects to make at least $2000 for each showing

• x+y≤300
  7.5x+5.5y≥2000

• x+y<300
  7.5x+5.5y≥2000

• x+y≤300
  7.5x+5.5y≤2000

• x+y>2000
  7.5x+5.5y≤300

JoAnn likes her job as a baby-sitter (b), but it pays only $3 per hour. She has been offered a job as a tutor (t) that pays $6 per hour.
Because of school, her parents only allow her to work a maximum of 15 hours per week, but she needs to make at least $66 per week. Which system of inequalities below correctly models Joann's situation?

• b + t < 15
  3b + 6t > 66

• b + t > 15
  3b + 6t < 66

• 3b + 6t > 15
  b + t < 66

• 3b + 6t < 15
  b + t < 66

Ms. Savva is ordering pizzas and breadsticks for a school party and has a budget of no more than $81. An order of breadsticks costs (x) $7 and a pizza costs (y) $13. 7x + 13y ≤ 81 Given Ms. Savva's budget, which of the following give a viable option she can purchase for the school party?

• 7 breadstick and 3 pizzas

• 3 breadstick and 4 pizzas