The students at Floyd Middle School have been working hard fundraising. They sold popcorn, had a bake sale and a car wash. Finally with the time for purchasing band uniforms rapidly approaching, Mrs. Kline gathered the whole band together one afternoon to discuss their profit.

“We did very well,” she started. “We raised a total of $12,000 and we had $1,000 in our account, so we have $13,000 to spend on our uniforms. I know that may seem like a lot of money, but uniforms are expensive. We are only going to be able to purchase new jackets for everyone. If we have any money left over, we’ll buy new fingerless gloves because some of the ones we’re using look awful.”

“Mrs. Kline, did you already pick the design of the jacket?” Kayla asked from the second row.

“Yes. It was one of the ones we voted on last year,” she said holding up a picture of the shiny, new navy jacket. “Now I need a few people to figure out the cost and if we have enough for the gloves too.”

Kayla and Juan volunteered to work on the arithmetic. Here is the information Mrs. Kline gave them.

The jackets each cost $99.95.

The total budget is $13,000.

There are 144 students in the band.

“We will need to spend $11,512.80 on the jackets,” Kayla said to Juan.

“Wow, that’s a lot of money. How much can we spend on the gloves?”

**That is a great question. It is one that can be answered by writing an inequality. The students need their total to be equal to or less than $13,000. In this Concept, you will learn how to work with inequalities that have addition and/or subtraction in them. Then you will use what you have learned to help Kayla and Juan with the band uniforms.**

### Guidance

Sometimes, you will have an inequality that is not as straightforward as \begin{align*}x>4\end{align*}. With this example, we know that the variable will be equal to any number that is greater than four. This is quite easy to work with and we can write a set of numbers to make this inequality a true statement.

**What if it isn’t that simple?**

Sometimes, you will see an inequality like this one.

\begin{align*}x+3>7\end{align*}

Here we need to figure out the set of numbers that will make this a true statement. We are looking for a number that when added to three is greater than seven.

**To figure this out, we will need to solve this inequality.**

Solving an inequality is similar to solving an equation. Here are some number properties that can help you solve inequalities.

**The** *addition property of inequality***states that if the same number is added to each side of an inequality, the sense of the inequality stays the same.** In other words, the inequality symbol does not change.

If \begin{align*}a>b\end{align*}, then \begin{align*}a+c>b+c\end{align*}. If \begin{align*}a \ge b\end{align*}, then \begin{align*}a+c \ge b+c\end{align*}.

If \begin{align*}a<b\end{align*}, then \begin{align*}a+c<b+c\end{align*}. If \begin{align*}a \le b\end{align*}, then \begin{align*}a+c \le b+c\end{align*}.

What about subtraction? Remember, subtracting a number, \begin{align*}c\end{align*}, is the same as adding its opposite, \begin{align*}-c\end{align*}. So, the addition property of inequality applies to subtraction as well. We can also state this as it’s own property.

**The** *subtraction property of inequality***states that if the same number is subtracted from each side of an inequality, the sense of the inequality stays the same.** In other words, the inequality symbol does not change.

If \begin{align*}a>b\end{align*}, then \begin{align*}a-c > b-c\end{align*}. If \begin{align*}a \ge b\end{align*}, then \begin{align*}a-c \ge b-c\end{align*}.

If \begin{align*}a<b\end{align*}, then \begin{align*}a-c < b-c\end{align*}. If \begin{align*}a \le b\end{align*}, then \begin{align*}a-c \le b-c\end{align*}.

Applying these properties makes our work quite simple. You can think of solving inequalities in the same way that you thought of solving equations. The big difference is that your answer will be a set of numbers and not a single number. Just like with equations, you need to be sure that your answer makes the mathematical statement true. If it doesn’t, then you need to rethink your answer.

**Now let’s look at solving inequalities.**

*Solve this inequality and graph its solution: \begin{align*}n-3<5\end{align*}.*

**Solve the inequality as you would solve an equation, by using inverse operations.** Since the 3 is subtracted from \begin{align*}n\end{align*}, add 3 to both sides of the inequality to solve it. Since you are adding the same number to both sides of the inequality, the addition property of inequality applies. According to that property, we know that the inequality symbol should stay the same when we add the same number, 3, to both sides of the inequality.

*Think back! To solve inequalities, you may need to remember how to add and subtract integers. Pay attention to the sign when you work with these values.*

\begin{align*}n-3 &< 5\\ n-3+3 &< 5+3\\ n+(-3+3) &< 8\\ n+0 &< 8\\ n &< 8\end{align*}

**Now, graph the solution. The inequality \begin{align*}n<8\end{align*} is read as “\begin{align*}n\end{align*} is less than 8.” So, the solution set for this inequality includes all numbers that are less than 8, but it does not include 8.**

Draw a number line from 0 to 10. Add an open circle at 8 to show that 7 is *not* a solution for this inequality. Then draw an arrow showing all numbers less than 8.

*Solve this inequality and graph its solution: \begin{align*}-2 \le x+4\end{align*}*

**Use inverse operations to isolate the variable.** Since the 4 is added to \begin{align*}x\end{align*}, subtract 4 from both sides of the inequality to solve it. Since you are subtracting the same number from both sides of the inequality, the subtraction property of inequality applies. According to that property, the inequality symbol should stay the same when we subtract the same number, 4, from both sides of the inequality.

\begin{align*}-2 & \le x+4\\ -2-4 & \le x+4-4\\ -2+(-4) &\le x+0\\ -6 & \le x\end{align*}

**Now, we should graph the solution.** However, before we can do that, we need to rewrite the inequality so the variable is listed first. The inequality \begin{align*}-6 \le x\end{align*} is read as “-6 is less than or equal to \begin{align*}x\end{align*}.” If we list the \begin{align*}x\end{align*} first, we must reverse the inequality symbol. That means changing the “less than or equal to” symbol \begin{align*}(\le)\end{align*} to a “less than or equal to symbol” \begin{align*}(\ge)\end{align*}.

So, \begin{align*}-6 \le x\end{align*} equivalent to \begin{align*}x \ge -6\end{align*}.

This makes sense. If -6 is less than or equal to \begin{align*}x\end{align*}, then \begin{align*}x\end{align*} must be greater than or equal to -6.

The inequality \begin{align*}x \ge -6\end{align*} is read as “\begin{align*}x\end{align*} is greater than or equal to -6.” So, the solutions of this inequality include -6 and all numbers that are greater than -6.

Draw a number line from -10 to 0. Add a closed circle at -6 to show that -6 *is* a solution for this inequality. Then draw an arrow showing all numbers greater than -6.

#### Example A

\begin{align*}x+5<-12\end{align*}

**Solution: \begin{align*}x<-17\end{align*}**

#### Example B

\begin{align*}y-8\le5\end{align*}

**Solution: \begin{align*}y\le13\end{align*}**

#### Example C

\begin{align*}a-5\ge22\end{align*}

**Solution: \begin{align*}a\ge27\end{align*}**

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

First, we write an inequality to represent the uniform cost, unknown cost of gloves, and the total budget for everything. The total of the uniforms and gloves must be less than or equal to the total budget.

\begin{align*}\$ 11,512 = \text{cost of uniforms}\end{align*}

\begin{align*}x = \text{budget for gloves}\end{align*}

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

$13,000

**Here is the inequality.**

\begin{align*}11,512 + x \le 13,000\end{align*}

**We solve the inequality by using inverse operations.**

\begin{align*}x &\le 13,000 - 11,512\\ x & \le \$1487.20\end{align*}

**The students will have a maximum budget of $1487.20 to spend on the gloves.**

### 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.

- Addition Property of Inequality
- You can add a quantity to both sides of an inequality and it does not change the sense of the inequality.

- Subtraction Property of Inequality
- You can subtract a quantity from both sides of an inequality and it does not change the sense of the inequality.

### Guided Practice

Here is one for you to try on your own.

At the store, Talia bought one item—a $4.99 bottle of shampoo. Let \begin{align*}d\end{align*} represent the amount in dollars that she handed the clerk. She received more than $5 in change.

a. Write an inequality to represent \begin{align*}d\end{align*}, the number of dollars Talia handed the clerk to pay for the shampoo.

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

**Solution**

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

Use a number, an operation sign, a variable, or an inequality symbol to represent each part of the problem. The fact that this problem involves “change” may help you see that you should write a subtraction expression to represent the first part of this problem. To represent how much change Talia received, you will need to subtract the cost of the shampoo from the amount she handed the clerk. The key words “more than”, in this case, indicate that you should use a \begin{align*} > \end{align*} symbol. Use this information to write an inequality for this problem.

\begin{align*}& (\text{dollars handed to the clerk}) - (\text{cost of \$4.99 bottle of shampoo}) > (\text{\$5 in change})\\ & \qquad \qquad \qquad \downarrow \qquad \qquad \qquad \downarrow \qquad \qquad \qquad \qquad \downarrow \qquad \qquad \qquad \quad \downarrow \qquad \quad \downarrow\\ & \qquad \qquad \qquad d \qquad \qquad \qquad - \qquad \qquad \qquad \quad 4.99 \qquad \qquad \qquad > \quad \quad \ \ 5\end{align*}

So, this problem can be represented by the inequality \begin{align*}d-4.99 > 5\end{align*}.

**Next, consider part** *b***.**

Solve the inequality to help you find three possible values for \begin{align*}d\end{align*}. To solve this inequality, add 4.99 to both sides. Do not change the inequality symbol.

\begin{align*}d-4.99 & > 5\\ d - 4.99+4.99 &> 5+4.99\\ d+(-4.99+4.99) &> 5.00+4.99\\ d+0 &> 9.99\\ d &> 9.99\end{align*}

According to the inequality above, the amount Talia handed the clerk was more than $9.99.

**So, three possible values of \begin{align*}d\end{align*} are $10.00, $10.99, and $20.00. These are only 3 possible answers. You could choose any amount that is greater than $9.99.**

### Video Review

Khan Academy Inequalities Using Addition and Subtraction

### Practice

Directions: Solve the following inequalities.

- \begin{align*}x+4>10\end{align*}
- \begin{align*}y-11<20\end{align*}
- \begin{align*}a+2<1\end{align*}
- \begin{align*}b+3 \ge 5\end{align*}
- \begin{align*}y-2 \le -4\end{align*}
- \begin{align*}x+1 \ge -5\end{align*}
- \begin{align*}x-3<11\end{align*}
- \begin{align*}x-4>-3\end{align*}
- \begin{align*}y+7>22\end{align*}
- \begin{align*}a-6\ge-1\end{align*}
- \begin{align*}b+14>20\end{align*}
- \begin{align*}x-24>-11\end{align*}
- \begin{align*}a+3\le-9\end{align*}
- \begin{align*}x-12>1\end{align*}
- \begin{align*}y+13>-33\end{align*}