Solutions to Systems of Linear Inequalities in Two Variables

ID: rital-kudof
Created by Illustrative MathIllustrative Math
Subject: Algebra, Algebra 2
Grade: 8-9

Solutions to Systems of Linear Inequalities in Two Variables

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Problem 1

1) Two inequalities are graphed on the same coordinate plane.

Which region represents the solution to the system of the two inequalities?

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a) Region A\text{Region A}b) Region B\text{Region B}c) Region C\text{Region C}d) Region D\text{Region D}
Problem 2

2) Select all the pairs of xx and yy that are solutions to the system of inequalities: {y2x+6xy<6\begin{cases} y \leq -2x + 6 \\ x - y < 6 \end{cases}

Write each corresponding letter in the answer box and separate letters with commas.

a) x=0x = 0, y=0y = 0 \quad \quad b) x=5x = -5, y=15y = -15 \quad \quad c) x=4x = 4, y=2y = -2 \quad \quad d) x=3x = 3, y=0y = 0 \quad \quad e) x=10x = 10, y=0y = 0

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Problem 3

Jada has $200 to spend on flowers for a school celebration. She decides that the only flowers that she wants to buy are roses, rr, and carnations, cc. Roses cost $1.45 each and carnations cost $0.65 each. Jada buys enough roses so that each of the 75 people attending the event can take home at least one rose.

3) Write an inequality to represent the constraint that every person takes home at least one rose.

4) Write an inequality to represent the cost constraint.

Problem 4

Here are the graphs of the equations 3x+y=93x + y = 9 and 3xy=93x - y = 9 on the same coordinate plane.

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5) Label each graph with the equation it represents.

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6) Identify the region that represents the solution set to 3x+y<93x + y < 9. Use a colored pencil or cross-hatching to shade the region.

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7) Is the boundary line a part of the solution?

True or false? Write below.

8) Identify the region that represents the solution set to 3xy<93x - y < 9. Use a different colored pencil or cross-hatching to shade the region.

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9) Is the boundary line a part of the solution?

True or false? Write below.

10) Identify a point that is a solution to both 3x+y<93x + y < 9 and 3xy<93x - y < 9.

Problem 5

11) Which coordinate pair is a solution to the inequality 4x2y<224x - 2y < 22?

a) (4, -3)\text{(4, -3)}b) (4, 3)\text{(4, 3)}c) (8, -3)\text{(8, -3)}d) (8, 3)\text{(8, 3)}
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Problem 6

Consider the linear equation 9x3y=129x - 3y = 12.

12) The pair (3, 5) is a solution to the equation. Find another pair (xx, yy) that is a solution to the equation.

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13) Is (3, 5) a solution to the inequality 9x3y129x - 3y \leq 12?

True or false? Write below.

14) Explain how you know.

15) Is (2, -10) a solution to the inequality 9x3y129x - 3y \leq 12?

True or false? Write below.

16) Explain how you know.

Problem 7

Elena is considering buying bracelets and necklaces as gifts for her friends. Bracelets cost $3, and necklaces cost $5. She can spend no more than $30 on the gifts.

17) Write an inequality to represent the number of bracelets, bb, and the number of necklaces nn, she could buy while sticking to her budget.

18) Graph the solutions to the inequality on the coordinate plane.

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19) Explain how we could check if the boundary is included or excluded from the solution set.

Problem 8

In physical education class, Mai takes 10 free throws, ff, and 10 jump shots, jj. She earns 1 point for each free throw she makes and 2 points for each jump shot she makes. The greatest number of points that she can earn is 30.

20) Write an inequality to describe the constraints.

21) Specify what each variable represents.

22) Name one solution to the inequality and explain what it represents in that situation.

Problem 9

A rectangle with a width of ww and a length of ll has a perimeter greater than 100.

Here is a graph that represents this situation.

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23) Write an inequality that represents this situation.

24) Can the rectangle have width of 45 and a length of 10?

True or false? Write below.

25) Explain your reasoning.

26) Can the rectangle have a width of 30 and a length of 20?

True or false? Write below.

27) Explain your reasoning.