# 7.14: One-Step Inequalities

**At Grade**Created by: CK-12

**Practice**One-Step Inequalities

Do you like to go to the movies? Take a look at this dilemma.

Marc and Kara have made several friends swimming at the town pool. One rainy day, they decided to invite the group to go to a movie. Everyone is very excited. Grandma is so pleased that Kara and Marc have been making friends that she has offered to treat.

“How many of you are going?”she asks Kara when she hears of the plan.

“Well, there are eight of us altogether,” Kara said.

“Alright, let me see,” Grandma says reaching into her wallet. “I have forty-eight dollars to contribute. Here it is.”

“Thank you Grandma,” the twins add smiling.

Later at the movie theater, Kara takes out the money. They have $48.00 to spend on tickets. Last minute, one of the friends has brought her brother along. Kara isn’t sure that they have enough money for the brother too. If each ticket is $6.00, how many tickets can she buy without going over the $48.00?

**This problem requires an inequality. The total must be less than or equal to $48.00 without going over. The movie is going to start soon. Pay attention to the work in this Concept and you will know how to help Kara purchase the tickets at the end of it.**

### Guidance

Now that you have developed an understanding of inequalities we can solve them and graph the solution sets.

We can solve an inequality in a similar way as we would use to solve an equation. The tricky part is in the answer not in the process. Once we solve the inequality, we can graph the solution.

Here is a problem with an inequality.

Solve this inequality and graph its solution \begin{align*}n-4 \le 3\end{align*}.

**Solve the inequality as you would solve an equation, by using inverse operations. Since the 4 is subtracted from \begin{align*}n\end{align*}, add 4 to both sides of the inequality to solve it.** You do not need to multiply or divide both sides by a negative number, so you do not need to reverse the inequality symbol. The symbol should stay the same.

\begin{align*}n-4 & \le 3\\ n-4 +4 & \le 3+4\\ n+(-4+4) & \le 7\\ n+0 & \le 7\\ n & \le 7\end{align*}

**Now, graph the solution.** The inequality \begin{align*}n \le 7\end{align*} is read as “\begin{align*}n\end{align*} is less than or equal to 7.” So, the solutions of this inequality include 7 and all numbers that are less than 7.

Draw a number line from 0 to 10. Add a closed circle at 7 to show that 7 is a solution for this inequality. Then draw an arrow showing all numbers less than 7.

**The solution for this inequality is \begin{align*}n \le 7\end{align*}, and its graph is shown above.**

Solve this inequality and graph its solution \begin{align*}-2n<14\end{align*}.

**Solve this inequality as you would solve an equation, by using inverse operations. Since the is multiplied by the \begin{align*}n\end{align*}, divide both sides of the inequality by to solve it.** Since this involves multiplying both sides of the inequality by a negative number, the sense of the inequality will change and you will need to reverse the inequality symbol. This means changing the inequality symbol from a “less than” symbol (<) to a “greater than” symbol (>).

\begin{align*}-2n & < 14\\ \frac{-2n}{-2} & > \frac{14}{-2}\\ 1n & > -7\\ n & > -7\end{align*}

**Now, graph the solution.** The inequality \begin{align*}n>-7\end{align*} is read as “\begin{align*}n\end{align*} is greater than or equal to 7.” So, the solutions of this inequality include all numbers that are greater than 7.

Draw a number line from –10 to 0. Add an open circle at -7 to show that -7 is not a solution for this inequality. Then draw an arrow showing all numbers greater than -7.

**The solution for this inequality is \begin{align*}n>-7\end{align*}, and its graph is shown above.**

Sometimes, you will need to take more than one step to solve an inequality. You can think of these problems in the same way that you thought about two-step equations.

\begin{align*}\frac{n}{3}+9 \ge -9\end{align*}.

**Solve this inequality as you would solve an equation, by using inverse operations. First, try to get the term with the variable, \begin{align*}\frac{n}{3}\end{align*}, by itself on one side of the inequality. Since the 9 is being added to \begin{align*}\frac{n}{3}\end{align*}, subtract 9 from both sides of the inequality.** You do not need to multiply or divide both sides by a negative number, so you do not need to reverse the inequality symbol during this step.

\begin{align*}\frac{n}{3}+9 & \ge -9\\ \frac{n}{3}+9-9 & \ge -9-9\\ \frac{n}{3}+0 & \ge (-9+-9)\\ \frac{n}{3} & \ge -18\end{align*}

**There is a second step you must take to find the solution. Since \begin{align*}n\end{align*} is divided by 3, you must multiply both sides of the inequality by 3 to find its solution.** This involves multiplying by a positive number, 3, so you do not need to reverse the inequality symbol. Be careful! It is true that you will need to multiply 3 by –18 to find the solution. However, since you are not multiplying both sides of the inequality by a negative number, you do not reverse the inequality symbol.

\begin{align*}\frac{n}{3} & \ge -18\\ \frac{n}{3} \times 3 & \ge -18 \times 3\\ \frac{n}{3} \times \frac{3}{1} & \ge -54\\ \frac{n}{\cancel{3}} \times \frac{\cancel{3}}{1} & \ge -54\\ \frac{n}{1} & \ge -54\\ n & \ge -54\end{align*}

**The solution for this inequality is \begin{align*}n \ge -54\end{align*}.**

Solve each inequality.

#### Example A

\begin{align*}x-4<10\end{align*}

**Solution:\begin{align*}x<14\end{align*}**

#### Example B

\begin{align*}2y+4 \ge 12\end{align*}

**Solution:\begin{align*}y\ge 4\end{align*}**

#### Example C

\begin{align*}-4x \le 16\end{align*}

**Solution:\begin{align*}x \le -4\end{align*}**

Here is the original problem once again. Use what you have learned about inequalities and about solving inequalities to help you with the movie tickets.

Marc and Kara have made several friends swimming at the town pool. One rainy day, they decided to invite the group to go to a movie. Everyone is very excited. Grandma is so pleased that Kara and Marc have been making friends that she has offered to treat.

“How many of you are going?”she asks Kara when she hears of the plan.

“Well, there are eight of us altogether,” Kara said.

“Alright, let me see,” Grandma says reaching into her wallet. “I have forty-eight dollars to contribute. Here it is.”

“Thank you Grandma,” the twins add smiling.

Later at the movie theater, Kara takes out the money. They have $48.00 to spend on tickets. Last minute, one of the friends has brought her brother along. Kara isn’t sure that they have enough money for the brother too. If each ticket is $6.00, how many tickets can she buy without going over the $48.00?

**To figure this out, we need to write an inequality. The original group consisted of 8 kids without the extra brother. Each ticket is $6.00. We can use this in our inequality. If we have money left over, then we will know whether or not we can pay for him too.**

**We need to be less than or equal to $48.00.**

**Here is the inequality.**

\begin{align*}6x \le 48\end{align*}

**Now we can solve the inequality. The \begin{align*}x\end{align*} represents the number of tickets that we can purchase if they are $6.00 each.**

\begin{align*}6x & \le 48\\ \frac{6x}{6} & \le 48\\ x & \le 8\end{align*}

**This shows that the kids can purchase less than or equal to 8 tickets with the $48.00. They don’t have enough to pay for the brother too.**

**Kara tells the group this information and they all chip in with their own money. All of the kids are able to attend the movie thanks to the generosity of the group.**

### Vocabulary

Here are the vocabulary words in this Concept.

- Equation
- a number sentence with an equal sign where the quantity on one side of the equals is the same as the quantity on the other side of the equals.

- Inequality
- a number sentence where one side is not necessarily equal to the other side. There are several possible answers that will make an inequality a true statement.

- Equivalent Inequalities
- Two inequalities that are written differently, but still express the same number relationships.

### Guided Practice

Here is one for you to try on your own.

Antonio is buying milk for a breakfast event. Each container of milk costs $3. At most, he can spend $12 on milk for the event.

a. Write an inequality to represent, \begin{align*}c\end{align*}, the number of containers of milk he can buy.

b. Could Antonio buy 4 containers of milk for the event? Explain.

**Answer**

Consider part a first.

Use a number, an operation sign, a variable, or an inequality symbol to represent each part of the problem. Since each container of milk costs $3, you can find the total cost, in dollars, of the milk he buys by multiplying 3 by the number of containers. The key words “at most” indicate that you should use a \begin{align*}\le\end{align*} symbol.

\begin{align*}& \underline{\text{Each container}\ldots \text{costs} \ \$3}. \ \underline{\text{At most}}, \ \text{he can spend} \ \underline{\$12}\ldots\\ & \qquad \qquad \qquad \quad \downarrow \qquad \qquad \qquad \downarrow \qquad \qquad \qquad \quad \quad \ \ \downarrow\\ & \qquad \qquad \quad \quad \ c \times 3 \qquad \qquad \quad \le \qquad \qquad \qquad \qquad 12\end{align*}

You may also want to consider that the value of \begin{align*}c\end{align*} must be an integer greater than or equal to 0. Think about why that is for a moment.

The reason that the value of \begin{align*}c\end{align*} must be an integer greater than or equal to zero is because Antonio cannot buy a negative number of containers nor can he buy a fraction of a container. Neither of those situations makes sense in real life. When using inequalities to represent real-life situations, you should always think about which values would make sense for the variable and which values would not make sense.

Next, consider part \begin{align*}b\end{align*}.

Solve the inequality to help you.

\begin{align*}c \times 3 & \le 12\\ \frac{c \times 3}{3} & \le \frac{12}{3}\\ c \times \frac{3}{3} & \le 4\\ c \times 1 & \le 4\\ c & \le 4\end{align*}

According to the inequality above, the number of containers, \begin{align*}c\end{align*}, that he can buy must be less than or equal to 4.

**Since 4 is a solution for this inequality, and since 4 is an integer that is greater than 0, he could buy 4 containers of milk.**

### Video Review

Here are videos for review.

### Practice

Directions: Solve each inequality.

1. \begin{align*}x+4<10\end{align*}

2. \begin{align*}x-3 \ge 7\end{align*}

3. \begin{align*}b+5 \le 15\end{align*}

4. \begin{align*}a-7 \ge 14\end{align*}

5. \begin{align*}4y>20\end{align*}

6. \begin{align*}6x \le 18\end{align*}

7. \begin{align*}-4y< -12\end{align*}

8. \begin{align*}-5x< -20\end{align*}

9. \begin{align*}\frac{x}{2}=10\end{align*}

10. \begin{align*}\frac{x}{5} \le 6\end{align*}

11. \begin{align*}2x+5 \ge 7\end{align*}

12. \begin{align*}3y-2 \le 4\end{align*}

13. \begin{align*}3a-7>11\end{align*}

14. \begin{align*}2b+9<39\end{align*}

Directions: Solve each problem.

15. Emma bought a fruit smoothie at a juice shop for \begin{align*}t\end{align*} dollars, including tax. Emma paid with a $10 bill. She received less than $5 in change.

a. Write an inequality to represent \begin{align*}t\end{align*}, the number of dollars, including tax, that Emma paid for the fruit smoothie.

b. List three possible values of \begin{align*}t\end{align*}.

16. Kiet has 16 juice boxes for a family picnic and needs to buy more. Juice boxes are sold in packages of 8.

a. Write an inequality to represent \begin{align*}p\end{align*}, the number of packages of juice boxes Kiet needs to buy in order to have at least 40 juice boxes total for the picnic.

b. If Kiet buys 4 packages of juice boxes, will that be enough?

Equivalent Inequalities

Equivalent inequalities are inequalities that can each be simplified to the same inequality.inequality

An inequality is a mathematical statement that relates expressions that are not necessarily equal by using an inequality symbol. The inequality symbols are , , , and .### Image Attributions

Here you'll learn to solve inequalities and graph solutions.

## Concept Nodes:

Equivalent Inequalities

Equivalent inequalities are inequalities that can each be simplified to the same inequality.inequality

An inequality is a mathematical statement that relates expressions that are not necessarily equal by using an inequality symbol. The inequality symbols are , , , and .