# Lesson 19

Solutions to Inequalities in One Variable

- Let’s find and interpret 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.

$x=\text-3$

$x=\text-2$

$x=\text-1$

$x=0$

$x=1$

$x=2$

$x=3$

### Problem 2

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

$x<\frac 12$

$x>\frac 12$

$x\leq \frac 12$

$x\geq \frac 12$

### 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 \(5+8x<3(2x+4)\) ?

### Problem 6

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

### Problem 7

Kiran has 27 nickels and quarters in his pocket, worth a total of \$2.75.

- 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.
- How many nickels and quarters are in Kiran’s pocket? Show your reasoning.

### Problem 8

How many solutions does this system of equations have? Explain how you know.

\(\begin{cases} y+\frac23 x = 4 \\ 2x=12-3y \\ \end{cases}\)

### 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.

$n \geq 16$

$n \geq 32$

$s<2n$

$s=2n$

$3n \leq 225$

$3s \leq 225$

### 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.