# Writing and Solving Inequalities in One Variable

# Writing and Solving Inequalities in One Variable

##### Problem 1

1) Solve $2x < 10$.

2) Explain how to find the solution set.

##### Problem 2

3) 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.

4) 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.

5) Solve the inequality.

##### Problem 4

6) Clare made an error when solving $-4x + 3 < 23$.

$\begin{array}{rcr} \\[-1em] -4x + 3 & < & 23 \\[-1em] \\ \\[-1em] -4x & < & 20 \\[-1em] \\ \\[-1em] x & < & -5 \\[-1em] \end{array}$

Describe the error that she made.

##### 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.

7) Write an inequality that expresses the mean number of steps, $m$, that Diego needs to walk during the last 3 days of this week to walk more than 70,000 steps.

8) Define $m$.

9) 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

10) 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?

##### Problem 7

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

##### Problem 8

Solve each system of equations without graphing.

12) $\begin{cases} 5x + 14y = -5 \\ -3x + 10y = 72 \end{cases}$

13) $\begin{cases} 20x - 5y = 289 \\ 22x + 9y = 257 \end{cases}$

##### 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:

$\begin{cases} 96x + 180y = 696 \\ 96x - 72y = 1,200 \end{cases}$

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."

14) Do you agree with Lin that her approach also works?

15) Explain your reasoning.

16) What are the smallest whole-number factors by which you can multiply the equations in order to eliminate $x$?

##### Problem 10

17) What is the solution set of the inequality $\frac{x+2}{2} \geq -7 - \frac{x}{2}$?