Lesson 21

Graphing Linear Inequalities in Two Variables (Part 1)

Problem 1

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

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1. Are the points $$(0,\frac12)$$ and $$(\text-7,\text-3)$$ solutions to the equation? Explain or show how you know.
2. Check if each of these points is a solution to the inequality $$2y -x > 1$$:

• $$(0,2)$$
• $$(8,\frac{1}{2})$$
• $$(\text{-}6,3)$$
• $$(\text{-}7,\text{-}3)$$
3. Shade the region that represents the solution set to the inequality $$2y -x > 1$$.

4. Are the points on the line included in the solution set? Explain how you know.

Problem 2

Select all coordinate pairs that are solutions to the inequality $$5x+9y< 45$$.

A:

$$(0,0)$$

B:

$$(5,0)$$

C:

$$(9,0)$$

D:

$$(0,5)$$

E:

$$(0,9)$$

F:

$$(5, 9)$$

G:

$$(\text-5,\text-9)$$

Problem 3

Consider the linear equation $$2y - 3x = 5$$

1. The pair $$(\text-1,1)$$ is a solution to the equation. Find another $$(x,y)$$ pair that is a solution to the equation.
2. Are $$(\text-1,1)$$ and $$(4,1)$$  solutions to the inequality $$2y - 3x < 5$$? Explain how you know.
3. Explain how to use the answers to the previous questions to graph the solution set to the inequality $$2y - 3x < 5$$.

Problem 4

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

Problem 5

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

A:

$$x-3y<5$$

B:

$$x-3y \leq 5$$

C:

$$x-3y>5$$

D:

$$x-3y \geq 5$$

Problem 6

Solve each system of equations without graphing.

1. $$\begin{cases} 4d+7e=68 \\ \text-4d-6e=\text-72\\ \end{cases}$$

2. $$\begin{cases} \frac14 x+y=1 \\ \frac32 x-y=\frac43 \\ \end{cases}$$

Solution

(From Unit 2, Lesson 14.)

Problem 7

Mai and Tyler are selling items to earn money for their elementary school. The school earns $$w$$ dollars for every wreath sold and $$p$$ 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: $$\begin{cases}14w + 3p = 70.50\\ 10w + 7p = 62.50 \end{cases}$$

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

Solution

(From Unit 2, Lesson 15.)

Problem 8

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 $$d$$, the total amount in dollars that Elena spends on supplies?

A:

$$d > 40$$

B:

$$d \geq 40$$

C:

$$d < 40$$

D:

$$d \leq 40$$

Solution

(From Unit 2, Lesson 18.)

Problem 9

Solve this inequality: $$\displaystyle \frac{x-4}3 \geq \frac{x+3}2$$

Solution

(From Unit 2, Lesson 19.)

Problem 10

Which graph represents the solution to $$\dfrac{4x-8}3\leq 2x-5$$ ?

A:
B:
C:
D:

Solution

Solve $$\text-x < 3$$. Explain how to find the solution set.