Have you ever had to work with a budget? A school event often has a budget. Take a look at this dilemma.

Ajay is buying decorations for a school dance. He will spend $10 on balloons and \begin{align*}x\end{align*} dollars on streamers. At most, he could spend $18 on decorations.

Write an inequality to represent \begin{align*}x\end{align*}, the number of dollars he could spend on streamers. This Concept will help you to learn how to do this.

### Guidance

We are going to begin working with **quantities that may or may not be equal.**

**These are called inequalities.**

Just like an equation is shown using an equal sign, an inequality is expressed through symbols too.

\begin{align*} > \end{align*} means “is greater than.”

\begin{align*} < \end{align*} means “is less than.”

\begin{align*}\ge\end{align*} means “is greater than or equal to.”

\begin{align*}\le\end{align*} means “is less than or equal to.”

*Take a minute and write the definition of an inequality and these symbols into your notebook.*

Let’s begin by taking a look at some inequalities.

\begin{align*}2 > x\end{align*}

Here we have an inequality where we know that two is greater than the quantity of \begin{align*}x\end{align*}. This problem has many different solutions. Any number that would make this a true statement can be substituted in for the variable. We know that we can have negative numbers here too. We can write our answer as a set of numbers that begins with one and goes down from there.

**The answer is {1, 0 , -1...}**

Here is another one.

\begin{align*}5 \le y\end{align*}

Here we have an inequality that involves less than or equal to. We can substitute any value in for the variable that makes this a true statement. This means that we can start with five and go greater.

**The answer is {5, 6, 7...}**

**We can also graph inequalities on a number line.** We graph inequalities on a number line to visually show the set of numbers that would make the statement a true statement. Given that we can have less than, greater than, or less than or equal to, and greater than or equal to, we can have four different types of graphing. Here are some hints to help you with graphing inequalities.

- Use an open circle to show that a value is
*not*a solution for the inequality. You will use open circles to graph inequalities that include the symbols \begin{align*} > \end{align*} or \begin{align*} < \end{align*}. - Use a closed circle to show that a value
*is*a solution for the inequality. You will use closed circles to graph inequalities that include the symbols \begin{align*}\ge\end{align*} or \begin{align*}\le\end{align*}.

*Write these hints in your notebook. Be sure to write the heading “Graphing Inequalities” with these hints.*

*Write an inequality to represent all possible values of \begin{align*}n\end{align*} if \begin{align*}n\end{align*} is less than 2. Then graph those solutions on a number line.*

**First, translate the description above into an inequality. You can do this in the same way that you wrote equations. Just pay close attention to the words being used.**

\begin{align*}& \ \underline{n}, \underline{is \ less \ than} \ \underline{2}.\\ & \downarrow \qquad \quad \downarrow \quad \ \ \downarrow\\ & \ n \qquad \ < \quad \ \ 2\end{align*}

**Now, graph the inequality.**

Draw a number line from -5 to 5.

**You know that \begin{align*}n\end{align*} represents all numbers less than 2, and can be represented as \begin{align*}n<2\end{align*}. So, the solutions for this inequality include all numbers less than 2. The number 2 is** *not***a solution for this inequality, so draw an open circle at 2. Then draw an arrow showing all numbers less than 2. The arrow should face left because the lesser numbers are to the left on a number line.**

Write an inequality for each example.

#### Example A

The quantities less than or equal to 4.

**Solution: \begin{align*}x\le4\end{align*}**

#### Example B

A number is greater than or equal to -12.

**Solution: \begin{align*}a\ge-12\end{align*}**

#### Example C

Two times a number is less than 7.

**Solution: \begin{align*}2x<7\end{align*}**

Now let's go back to the dilemma at the beginning of the Concept.

**Consider part** *a***first.**

Use a number, an operation sign, a variable, or an inequality symbol to represent each part of the problem. The key words “at most” indicate that you should use a \begin{align*}\le\end{align*} symbol.

\begin{align*}& \ \underline{\$10 \ on \ balloons} \ \underline{and} \ \underline{x \ dollars \ on \ streamers} \ \ldots \ \underline{At \ most}, \ he \ can \ spend \ \underline{\$18} \ldots \\ & \qquad \quad \ \downarrow \qquad \qquad \downarrow \qquad \qquad \quad \ \ \downarrow \qquad \qquad \qquad \qquad \downarrow \qquad \qquad \qquad \qquad \downarrow\\ & \qquad \quad \ 10 \qquad \quad \ + \qquad \qquad \quad \ x \qquad \quad \qquad \qquad \ \ \le \qquad \quad \qquad \qquad \ 18\end{align*}

**So, the inequality \begin{align*}10+x \le 18\end{align*} represents \begin{align*}x\end{align*}, the number of dollars Ajay spent on streamers.**

### Vocabulary

- Equation
- a mathematical statement using an equals sign where the quantity on one side of the equals is the same as the quantity on the other side.

- Inequality
- a mathematical statement where the value on one side of an inequality symbol can be less than, greater than and sometimes also equal to the quantity on the other side. The key is that the quantities are not necessarily equal.

### Guided Practice

Here is one for you to try on your own.

Write an inequality to represent all possible values of \begin{align*}n\end{align*} if \begin{align*}n\end{align*} is greater than or equal to -4. Then graph those solutions on a number line.

**Solution**

First, translate the description above into an inequality.

\begin{align*}& \ \underline{n} \ \underline{is \ greater \ than \ or \ equal \ to} \ \underline{-4}.\\ & \downarrow \qquad \qquad \quad \ \ \downarrow \qquad \qquad \qquad \ \downarrow\\ & \ n \qquad \qquad \quad \ \ge \qquad \qquad \quad \ \ -4\end{align*}

**Now, graph the inequality.**

**Draw a number line from -5 to 5.**

**You know that \begin{align*}n\end{align*} represents all numbers greater than or equal to -4, and can be represented as \begin{align*}n \ge -4\end{align*}. So, the solutions for this inequality include all numbers greater than or equal to -4. The number -4 is a solution for this inequality, so draw a closed circle at -4. Then draw an arrow showing all numbers greater than -4. The arrow should face right because the greater numbers are to the right on a number line.**

### Video Review

Khan Academy Graph Inequalities on a Number line

### Practice

Directions: Write a solution set for each inequality. Include at least three values in your solution set.

- \begin{align*}x<13\end{align*}
- \begin{align*}y>5\end{align*}
- \begin{align*}x<2\end{align*}
- \begin{align*}y> -3\end{align*}
- \begin{align*}a>12\end{align*}
- \begin{align*}x \le 4\end{align*}
- \begin{align*}y \ge 3\end{align*}
- \begin{align*}b \ge -3\end{align*}
- \begin{align*}a \le -5\end{align*}
- \begin{align*}b \ge 11\end{align*}

Directions: Write an inequality to describe each situation.

- A number is less than or equal to -8.
- A number is greater than 50.
- A number is less than -4.
- A number is greater than -12.
- A number is greater than or equal to 11.