Modeling with Systems of Inequalities in Two Variables

ID: bijut-sujof
Created by Illustrative MathIllustrative Math
Subject: Algebra, Algebra 2
Grade: 8-9

Modeling with Systems of Inequalities in Two Variables

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Problem 1

The organizers of a conference needs to prepare at least 200 notepads for the event and have a budget of $160 for the notepads. A store sells notepads in packages of 24 and packages of 6.

This system of inequalities represent these constraints: {24x+6y20016x+5.40y160\begin{cases} 24x + 6y \geq 200 \\ 16x + 5.40y \leq 160 \end{cases}

1) Explain what the second inequality in the system tells us about the situation.

Here are incomplete graphs of the inequalities in the system, showing only the boundary lines of the solution regions.

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2) Which graph represents the boundary line of the second inequality?

3) Complete the graphs to show the solution set to the system of inequalities.

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4) Find a possible combination of large and small packages of notepads the organizer could order.

Problem 2

A certain stylist charges $15 for a haircut, xx, and $30 for hair coloring, yy. A haircut takes on average 30 minutes, while coloring takes 2 hours. The stylist works up to 8 hours in a day, and she needs to make a minimum of $150 a day to pay for her expenses.

5) Create a system of inequalities that describes the constraints in this situation.

6) Specify what each variable represents.

7) Graph the inequalities and show the solution set.

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8) Identify a point that represents a combination of haircuts and and hair-coloring jobs that meets the stylist’s requirements.

9) Identify a point that is a solution to the system of inequalities but is not possible or not likely in the situation.

10) Explain why this solution is impossible or unlikely.

Problem 3

11) Choose the graph that shows the solution to this system: {y>3x+24x+3y12\begin{cases} y > 3x + 2 \\ -4x + 3y \leq 12 \end{cases}

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a) Graph A\text{Graph A}b) Graph B\text{Graph B}c) Graph C\text{Graph C}d) Graph D\text{Graph D}
Problem 4

Match each inequality to the graph of its solution. Write the letter of the correct graph in the answer box.

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12) 2x5y202x - 5y \geq 20

13) 5x+2y205x + 2y \geq 20

14) 4x10y204x - 10y \leq 20

15) 4x5y204x - 5y \geq 20

16) 2x+10y202x + 10y \leq 20