Lesson 20

Writing and Solving Inequalities in One Variable

  • Let’s solve problems by writing and solving inequalities in one variable.

Problem 1

Solve \(2x < 10\). Explain how to find the solution set.

Problem 2

Lin is solving the inequality \(15 - x < 14\). She knows the solution to the equation \(15 - x = 14\) is \(x = 1\)

How can Lin determine whether \(x > 1\) or \(x < 1\) is the solution to the inequality?

Problem 3

A cell phone company offers two texting plans. People who use plan A pay 10 cents for each text sent or received. People who use plan B pay 12 dollars per month, and then pay an additional 2 cents for each text sent or received.

  1. Write an inequality to represent the fact that it is cheaper for someone to use plan A than plan B. Use \(x\) to represent the number of texts they send.

  2. Solve the inequality.

Problem 4

Clare made an error when solving \(\text-4x+3<23\).

Describe the error that she made.

\(\displaystyle \begin{align} \text-4x+3<23 \\ \text-4x<20 \\ x< \text-5 \end{align}\)


Problem 5

Diego’s goal is to walk more than 70,000 steps this week. The mean number of steps that Diego walked during the first 4 days of this week is 8,019.

  1. Write an inequality that expresses the mean number of steps that Diego needs to walk during the last 3 days of this week to walk more than 70,000 steps. Remember to define any variables that you use.

  2. If the mean number of steps Diego walks during the last 3 days of the week is 12,642, will Diego reach his goal of walking more that 70,000 steps this week?

Problem 6

Here are statistics for the length of some frog jumps in inches:

  • the mean is 41 inches
  • the median is 39 inches
  • the standard deviation is about 9.6 inches
  • the IQR is 5.5 inches

How does each statistic change if the length of the jumps are measured in feet instead of inches?

(From Unit 1, Lesson 15.)

Problem 7

Solve this system of linear equations without graphing: \(\begin{cases} 3y+7=5x \\ 7x-3y=1 \\ \end{cases}\)

(From Unit 2, Lesson 15.)

Problem 8

Solve each system of equations without graphing.

  1. \(\begin{cases} 5x+14y=\text-5 \\ \text-3x+10y=72 \\ \end{cases}\)

  2. \(\begin{cases}20x-5y=289 \\ 22x + 9y=257 \\ \end{cases}\)

(From Unit 2, Lesson 16.)

Problem 9

Noah and Lin are solving this system: \(\begin{cases} 8x+15y=58 \\ 12x-9y=150  \end{cases}\)

Noah multiplies the first equation by 12 and the second equation by 8, which gives:

\(\displaystyle \begin{cases} 96x+180y=696 \\ 96x-72y=1,\!200 \\ \end{cases}\)

Lin says, “I know you can eliminate \(x\) by doing that and then subtracting the second equation from the first, but I can use smaller numbers. Instead of what you did, try multiplying the first equation by 6 and the second equation by 4."

  1. Do you agree with Lin that her approach also works? Explain your reasoning.
  2. What are the smallest whole-number factors by which you can multiply the equations in order to eliminate \(x\)?

(From Unit 2, Lesson 16.)

Problem 10

What is the solution set of the inequality \(\dfrac{x+2}{2}\geq \text-7-\dfrac {x}{2}\) ?


\(x\leq \text-8\)


\(x\geq \text-8\)


\(x \geq - \frac92\)


\(x\geq 8\)

(From Unit 2, Lesson 19.)