# 3.16: Solve Inequalities Involving Combining Like Terms

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

**Practice**Inequalities with Like Terms

Your community center is selling boxes of cards. Your goal is to sell at least $500 worth of cards. Each box sells for $5 but you also need to pay the company $50 to ship all of the cards to your house to begin selling. Determine the least number of boxes of cards you must sell in order to meet your goal.

In this concept, you will learn to solve inequalities involving combining like terms.

### Multi-Step Inequalities

You can solve inequalities in many ways. Some inequalities can be solved in a single step. You could solve

However, two or more steps may be required to solve some inequalities. Inequalities that need more than one inverse operation to solve them can be called **multi-step inequalities**.

Let’s start by looking at combining like terms when you solve an inequality.

First, you can see that you have two terms that have the same variable. These are **like terms**. To solve an inequality with like terms, you will need to combine the like terms and then you can solve the inequality.

Here you divide both sides of the inequality by 7. Multiplication is the inverse operation of division.

Let’s look at another example.

Solve for

First, notice that there are two terms on the left side of the inequality, *added* to

The answer is

### Examples

#### Example 1

Earlier, you were given a problem about boxes of cards being sold.

First, write an inequality using the given information. You want to raise at least $500 by selling boxes of cards worth $5 each. You also need to pay the supplier $50 for shipping. Let ‘

Next, isolate the term with the variable,

Next, since

The answer is

You need to sell at least 110 boxes of cards.

#### Example 2

Solve for

First, subtract

Next, isolate the term with the variable,

Then, since

The answer is

#### Example 3

Solve for

First, isolate the term with the variable,

** 4x+54x+5−54x<<<2121−516 **

Next, since

The answer is \begin{align*}x < 4\end{align*}.

#### Example 4

Solve for \begin{align*}x\end{align*}:

\begin{align*}3x-6>30\end{align*}

First, isolate the term with the variable, \begin{align*}3x\end{align*}, on one side of the inequality. Since 6 is subtracted from \begin{align*}3x\end{align*}, you should add 6 to both sides of the inequality.

\begin{align*}\begin{array}{rcl}
3x -6&>& 30 \\
3x-6+6 &>& 30+6\\
3x&>& 36
\end{array}\end{align*}

Next, since \begin{align*}3x\end{align*} means \begin{align*}3 \times x\end{align*}, you can divide each side of the inequality by 3 to get ‘\begin{align*}x\end{align*}’ by itself on one side of the inequality.

\begin{align*}\begin{array}{rcl} 3x &>&36 \\ \frac{3x}{3} &>& \frac{36}{3}\\ x&>& 12 \end{array}\end{align*}

.\begin{align*}x>12\end{align*}The answer is

#### Example 5

Solve for \begin{align*}a\end{align*}:

\begin{align*}-3a+2<14\end{align*}

First, isolate the term with the variable, \begin{align*}-3a\end{align*}, on one side of the inequality. Since 2 is added to \begin{align*}-3a\end{align*}, you should subtract 2 from both sides of the inequality.

\begin{align*}\begin{array}{rcl}
-3a+2&<& 14 \\
-3a+2-2 &<& 14-2\\
-3a&<& 12
\end{array}\end{align*}

Next, since\begin{align*}-3a\end{align*} means \begin{align*}-3 × a\end{align*}, you can divide each side of the inequality by -3 to get ‘\begin{align*}a\end{align*}’ by itself on one side of the equation. Remember to reverse the inequality sign since you are dividing by a negative number.

\begin{align*}\begin{array}{rcl} -3a &<& 12 \\ \frac{-3a}{-3} &<& \frac{12}{-3}\\ a&>& -4 \end{array}\end{align*} The answer is \begin{align*}a > -4\end{align*}.

### Review

Solve each inequality.

1. \begin{align*}2x+5>13\end{align*}

2. \begin{align*}4x-2<10\end{align*}

3.\begin{align*}6y+9>69\end{align*}

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

5. \begin{align*}5x+2 \ge -8\end{align*}

6. \begin{align*}2x-9 \le -5\end{align*}

7. \begin{align*}\frac{x}{3}+1>5\end{align*}

8. \begin{align*}\frac{x}{2}-1<3\end{align*}

9. \begin{align*}\frac{x}{5}+3>-9\end{align*}

10. \begin{align*}\frac{x}{2}-5>-10\end{align*}

11. \begin{align*}6k-3>15\end{align*}

12. \begin{align*}11+\frac{x}{4}\le 12\end{align*}

13. \begin{align*}12+9j+j<72\end{align*}

14. \begin{align*}12b-3b+5 \ge -31\end{align*}

15. \begin{align*}18+7n+3+6n \le 86\end{align*}

16. \begin{align*}3z-15z-30>54\end{align*}

### Review (Answers)

To see the Review answers, open this PDF file and look for section 3.16.

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 .Inverse Operation

Inverse operations are operations that "undo" each other. Multiplication is the inverse operation of division. Addition is the inverse operation of subtraction.like terms

Terms are considered like terms if they are composed of the same variables with the same exponents on each variable.### Image Attributions

In this concept, you will learn to solve inequalities involving combining like terms.

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