# Lesson 19

Solutions to Inequalities in One Variable

### Problem 1

Here is an inequality: $$\dfrac{7x+6}2 \leq 3x+2$$

Select all of the values that are a solution to the inequality.

A:

$$x=\text-3$$

B:

$$x=\text-2$$

C:

$$x=\text-1$$

D:

$$x=0$$

E:

$$x=1$$

F:

$$x=2$$

G:

$$x=3$$

### Problem 2

Find the solution set to this inequality: $$2x-3>\dfrac{2x-5}2$$

A:

$$x<\frac{1}{2}$$

B:

$$x>\frac 12$$

C:

$$x\leq \frac 12$$

D:

$$x\geq \frac 12$$

### Problem 3

Here is an inequality: $$\dfrac{\text-10+x}4+5 \geq \dfrac{7x-5}3$$

What value of $$x$$ will produce equality (or make the two sides equal)?

### Problem 4

Noah is solving the inequality $$7x + 5 > 2x + 35$$. First, he solves the equation $$7x + 5 = 2x + 35$$ and gets $$x = 6$$.

How does the solution to the equation $$7x + 5 = 2x + 35$$ help Noah solve the inequality $$7x + 5 > 2x + 35$$? Explain your reasoning.

### Problem 5

Which graph represents the solution to $$5+8x<3(2x+4)$$ ?

A:
B:
C:
D:

### Problem 6

Solve this system of linear equations without graphing: $$\begin{cases} 7x + 11y = \text-2 \\ 7x + 3y = 30 \end{cases}$$

### Solution

(From Unit 2, Lesson 15.)

Kiran has 27 nickels and quarters in his pocket, worth a total of \$2.75. 1. Write a system of equations to represent the relationships between the number of nickels $$n$$, the number of quarters $$q$$, and the dollar amount in this situation. 2. How many nickels and quarters are in Kiran’s pocket? Show your reasoning. ### Solution Teachers with a valid work email address can click here to register or sign in for free access to Formatted Solution. (From Unit 2, Lesson 12.) ### Problem 8 How many solutions does this system of equations have? Explain how you know. $$\begin{cases} y+\frac23 x = 4 \\ 2x=12-3y \\ \end{cases}$$ ### Solution Teachers with a valid work email address can click here to register or sign in for free access to Formatted Solution. (From Unit 2, Lesson 17.) ### Problem 9 The principal of a school is hosting a small luncheon for her staff. She plans to prepare two sandwiches for each person. Some staff members offer to bring salads and beverages. The principal has a budget of$225 and expects at least 16 people to attend. Sandwiches cost $3 each. Select all of the equations and inequalities that could represent the constraints in the situation, where $$n$$ is number of people attending and $$s$$ is number of sandwiches. A: $$n \geq 16$$ B: $$n \geq 32$$ C: $$s<2n$$ D: $$s=2n$$ E: $$3n \leq 225$$ F: $$3s \leq 225$$ ### Solution Teachers with a valid work email address can click here to register or sign in for free access to Formatted Solution. (From Unit 2, Lesson 18.) ### Problem 10 Students at the college are allowed to work on campus no more than 20 hours per week. The jobs that are available pay different rates, starting from \$8.75 an hour. Students can earn a maximum of \\$320 per week.

Write at least two inequalities that could represent the constraints in this situation. Be sure to specify what your variables represent.