Representing Situations with Inequalities

ID: lugib-bojoh
Created by Illustrative MathematicsIllustrative Mathematics, CC BY 4.0
Subject: Algebra, Algebra 2
Grade: 8-9
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18 questions

Representing Situations with Inequalities

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Problem 1

1) Tyler goes to the store. His budget is $125. Which inequality represents xx, the amount in dollars Tyler can spend at the store?









Problem 2

2) Jada is making lemonade for a get-together with her friends. She expects a total of 5 to 8 people to be there (including herself). She plans to prepare 2 cups of lemonade for each person.

The lemonade recipe calls for 4 scoops of lemonade powder for each quart of water. Each quart is equivalent to 4 cups.

Let nn represent the number of people at the get-together, cc the number of cups of water, ll the number of scoops of lemonade powder.

Select all the mathematical statements that represent the quantities and constraints in the situation. Write each corresponding letter in the answer box and separate letters with commas.

a) 5<n<85 < n < 8 \quad \quad  b) 5n85 \leq n \leq 8 \quad \quad c) c=2nc = 2n \quad \quad d) l=cl = c \quad \quad  e) 10<c<1610 < c < 16 \quad \quad f) 10l1610 \leq l \leq 16

Problem 3

A doctor sees between 7 and 12 patients each day. On Mondays and Tuesdays, the appointment times are 15 minutes. On Wednesdays and Thursdays, they are 30 minutes. On Fridays, they are one hour long. The doctor works for no more than 8 hours a day.

Here are some inequalities that represent this situation.

0.25y17x12xy80.25 \leq y \leq 1 \quad \quad \quad 7 \leq x \leq 12 \quad \quad \quad xy \leq 8

3) What does each variable represent?

4) What does the expression xyxy in the last inequality mean in this situation?

Problem 4

Han wants to build a dog house. He makes a list of the materials needed:

  • At least 60 square feet of plywood for the surfaces
  • At least 36 feet of wood planks for the frame of the dog house
  • Between 1 and 2 quarts of paint

Han's budget, bb, is $65. Plywood, pp, costs $0.70 per square foot. Planks of wood, ww, cost $0.10 per foot. Paint, qq, costs $8 per quart.

Write inequalities to represent the material constraints and cost constraints in this situation.

5) Write an inequality to represent the amount of plywood necessary to build the dog house.

6) Write an inequality to represent the number of wood planks necessary.

7) Write an inequality to represent the amount of paint necessary.

8) Write an inequality to represent the maximum budget.

9) Write an inequality to represent the relation between maximum budget and costs of materials.

10) Specify what the variables represent.

Problem 5

11) The equation V=13πr2hV = \frac{1}{3} \pi r^2 h represents the volume of a cone, where rr is the radius of the cone and hh is the height of the cone.

Which equation is solved for the height of the cone?


hh=VV-πr2\pi r^{2}


hh=13πr2V\frac{1}{3}\pi r^{2}V


3VV-πr2\pi r^{2}=hh


hh=3Vπr2\frac{3V}{\pi r^{2}}

Problem 6

Solve each system of equations without graphing.

12) {2x+3y=52x+4y=9\begin{cases} 2x + 3y = 5 \\ 2x + 4y = 9 \end{cases}

Show Work

13) {23x+y=7323xy=1\begin{cases} \frac{2}{3}x + y = \frac{7}{3} \\ \frac{2}{3}x - y = 1 \end{cases}

Show Work
Problem 7

14) There is a pair of xx and yy values that make each equation true in this system of equations:

{5x+3y=84x+7y=34\begin{cases} 5x + 3y = 8 \\ 4x + 7y = 34 \end{cases}

Explain why the same pair of values also make 9x+10y=429x + 10y = 42 true.

Problem 8

15) Which ordered pair is a solution to this system of equations? {7x+5y=593x9y=159\begin{cases} 7x + 5y = 59 \\ 3x - 9y = 159 \end{cases}


(-17, -12)


(-17, 12)


(17, -12)


(17, 12)

Show Work
Problem 9

16) Which equation has exactly one solution in common with the equation y=6x2y = 6x - 2?









Show Work
Problem 10

17) How many solutions does this system of equations have?

{y=4x+32x+8y=10\begin{cases} y = -4x + 3 \\ 2x + 8y = 10 \end{cases}

18) Explain how you know.