Graphing Linear Inequalities in Two Variables (Part 1)

ID: kazob-nofiv
Created by Illustrative MathIllustrative Math
Subject: Algebra, Algebra 2
Grade: 8-9

Graphing Linear Inequalities in Two Variables (Part 1)

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Problem 1

Here is a graph of the equation 2yx=12y - x = 1.

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1) Is the point (0, 12\frac{1}{2}) a solution to the equation?

True or false? Write below.

2) Explain or show how you know.

3) Is the point (-7, -3) a solution to the equation?

True or false? Write below.

4) Explain or show how you know.

Check if each of these points is a solution to the inequality 2yx>12y - x > 1:

5) (0, 2)

True or false? Write below.

6) (8, 12\frac{1}{2})

True or false? Write below.

7) (-6, 3)

True or false? Write below.

8) (-7, -3)

True or false? Write below.

9) Shade the region that represents the solution set to the inequality 2yx>12y - x > 1.

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10) Are the points on the line included in the solution set?

True or false? Write below.

11) Explain how you know.

Problem 2

12) Select all coordinate pairs that are solutions to the inequality 5x+9y<455x + 9y < 45. Write each corresponding letter in the answer box and separate letters with commas.

a) (0, 0) b) (5, 0) c) (9, 0) d) (0, 5) e) (0, 9) f) (5, 9) g) (-5, -9)

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Problem 3

Consider the linear equation 2y3x=52y - 3x = 5.

13) The pair (-1, 1) is a solution to the equation. Find another (xx, yy) pair that is a solution to the equation.

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14) Is (-1, 1) a solutions to the inequality 2y3x<52y - 3x < 5?

True or false? Write below.

15) Explain how you know.

16) Is (4, 1) a solution to the inequality 2y3x<52y - 3x < 5?

True or false? Write below.

17) Explain how you know.

18) Explain how to use the answers to the previous questions to graph the solution set to the inequality 2y3x<52y - 3x < 5.

Problem 4

19) The boundary line on the graph represents the equation 5x+2y=65x + 2y = 6. Write an inequality that is represented by the graph.

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Problem 5

20) Choose the inequality whose solution set is represented by this graph.

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a) x3y<5 x - 3y < 5 b) x3y5 x - 3y \leq 5 c) x3y>5 x - 3y > 5 d) x3y5 x - 3y \geq 5
Problem 6

Solve each system of equations without graphing.

21) {4d+7e=684d6e=72\begin{cases} 4d + 7e = 68 \\ -4d - 6e = -72 \end{cases}

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22) {14x+y=132xy=43\begin{cases} \frac{1}{4}x + y = 1 \\ \frac{3}{2}x - y = \frac{4}{3} \end{cases}

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Problem 7

23) Mai and Tyler are selling items to earn money for their elementary school. The school earns ww dollars for every wreath sold and pp dollars for every potted plant sold. Mai sells 14 wreaths and 3 potted plants and the school earns $70.50. Tyler sells 10 wreaths and 7 potted plants and the school earns $62.50.

This situation is represented by this system of equations: {14w+3p=70.5010w+7p=62.50\begin{cases} 14w + 3p = 70.50 \\ 10w + 7p = 62.50 \end{cases}

Explain why it makes sense in this situation that the solution of this system is also a solution to 4w+(4p)=8.004w + (-4p) = 8.00.

Problem 8

24) Elena is planning to go camping for the weekend and has already spent $40 on supplies. She goes to the store and buys more supplies.

Which inequality represents dd, the total amount in dollars that Elena spends on supplies?

a) d>40 d > 40 b) d40 d \geq 40 c) d<40 d < 40 d) d40 d \leq 40
Problem 9

25) Solve this inequality: x43x+32\frac{x-4}{3} \geq \frac{x + 3}{2}

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Problem 10

26) Which graph represents the solution to 4x832x5\frac{4x-8}{3} \leq 2x - 5?

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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 11

27) Solve x<3-x < 3.

28) Explain how to find the solution set.