Solving Problems with Inequalities in Two Variables

ID: vagon-valuj
Created by Illustrative MathIllustrative Math
Subject: Algebra, Algebra 2
Grade: 8-9

Solving Problems with Inequalities in Two Variables

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Problem 1

1) This year, students in the 9th grade are collecting dimes and quarters for a school fundraiser. They are trying to collect more money than the students who were in the 9th grade last year. The students in 9th grade last year collected $143.88.

Using dd to represent the number of dimes collected and qq to represent the number of quarters, which statement best represents this situation?

a) 0.25d+0.1q143.88 0.25d + 0.1q \geq 143.88 b) 0.25q+0.1d143.88 0.25q + 0.1d \geq 143.88 c) 0.25d+0.1q>143.88 0.25d + 0.1q > 143.88 d) 0.25q+0.1d>143.88 0.25q + 0.1d > 143.88
Problem 2

A farmer is creating a budget, bb, for planting soybeans and wheat. Planting soybeans, ss, costs $200 per acre and planting wheat, ww, costs $500 per acre. He wants to spend no more than $100,000 planting soybeans and wheat.

2) Write an inequality to describe the constraints.

3) Specify what each variable represents.

4) Name one solution to the inequality.

5) Explain what the solution represents in that situation.

Problem 3

Elena is ordering dried chili peppers and corn husks for her cooking class. Chili peppers cost $16.95 per pound and corn husks cost $6.49 per pound.

Elena spends less than $50 on dd pounds of dried chili peppers and hh pounds of corn husks.

Here is a graph that represents this situation.

A template for answering this question. Ask your instructor for an alternative.

6) Write an inequality that represents this situation.

7) Specify what each variable represents.

8) Can Elena purchase 2 pounds of dried chili peppers and 4 pounds of corn husks and spend less than $50?

True or false? Write below.

9) Explain your reasoning.

10) Can Elena purchase 1.5 pounds of dried chili peppers and 3 pounds of corn husks and spend less than $50?

True or false? Write below.

11) Explain your reasoning.

Problem 4

12) Which inequality is represented by the graph?

A template for answering this question. Ask your instructor for an alternative.
a) 4x2y>12 4x - 2y > 12 b) 4x2y<12 4x - 2y < 12 c) 4x+2y>12 4x + 2y > 12 d) 4x+2y<12 4x + 2y < 12
Problem 5

Here are some statistics for the number of minutes it took each of 9 members of the track team to run 1 mile.

  • mean: 7.3
  • median: 7.1
  • standard deviation: 1.1
  • Q1: 6.8
  • Q3: 7.4

When a tenth time is added to the list, the standard deviation increases to 1.5.

13) Is the tenth time likely an outlier?

True or false? Write below.

14) Explain your reasoning.

Problem 6

Elena is solving this system of equations: {10x6y=165x3y=8\begin{cases} 10x - 6y = 16 \\ 5x - 3y = 8 \end{cases}

She multiplies the second equation by 2, then subtracts the resulting equation from the first. To her surprise, she gets the equation 0 = 0.

15) What is special about this system of equations?

16) Why does she get this result and what does it mean about the solutions? (If you are not sure, try graphing them.)

Problem 7

Jada has a sleeping bag that is rated for 30^{\circ} F. This means that if the temperature outside, TT, is at least 30^{\circ} F, Jada will be able to stay warm in her sleeping bag.

17) Write an inequality that represents the outdoor temperature at which Jada will be able to stay warm in her sleeping bag.

18) Write an inequality that represents the outdoor temperature at which a thicker or warmer sleeping bag would be needed to keep Jada warm.

Problem 8

19) What is the solution set to this inequality:

6x20>3(2x)+6x26x - 20 > 3(2 - x) + 6x - 2?

Problem 9

Here is a graph of the equation 2x3y=152x - 3y = 15.

A template for answering this question. Ask your instructor for an alternative.

20) Is the point (1.5, -4) a solution to the equation?

True or false? Write below.

21) Is the point (4, -4) a solution to the equation?

True or false? Write below.

22) Explain or show how you know.

Check if each of these points is a solution to the inequality 2x3y<152x - 3y < 15:

23) (0,-5)

True or false? Write below.
Show Work

24) (4, -2)

True or false? Write below.
Show Work

25) (2, -4)

True or false? Write below.
Show Work

26) (5, -1)

True or false? Write below.
Show Work

27) Shade the solutions to the inequality .

A template for answering this question. Ask your instructor for an alternative.

28) Are the points on the line included in the solution region?

True or false? Write below.

29) Explain how you know.

Problem 10

A store sells notepads in packages of 24, ll, and packages of 6, ss. The organizers of a conference needs to prepare at least 200 notepads for the event.

Would they have enough notepads if they bought these quantities?

30) Seven packages of 24 and one package of 6

True or false? Write below.
Show Work

31) Five packages of 24 and fifteen packages of 6

True or false? Write below.
Show Work

32) Write an inequality to represent the relationship between the number of large and small packages of notepads and the number of notepads needed for the event.

33) Specify what each variable represents.

34) Use graphing technology to graph the solution set to the inequality. Then, use the graph to name two other possible combinations of large and small packages of notepads that will meet the number of notepads needed for the event.