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# 3.16: Solve Inequalities Involving Combining Like Terms

Difficulty Level: At Grade Created by: CK-12
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Practice Inequalities with Like Terms
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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. ### Guidance You can solve inequalities in many ways. Some inequalities can be solved in a single step. You could solve b+4<10\begin{align*}b + 4 < 10\end{align*} in one step—by subtracting 4 from each side. 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. 4x+3x<21\begin{align*}4x + 3x < 21\end{align*} 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. 7x<21\begin{align*}7x < 21\end{align*} Here you divide both sides of the inequality by 7. Multiplication is the inverse operation of division. 7x7x7x<<<212173 The answer is x<3\begin{align*}x<3\end{align*}. Let’s look at another example. Solve for b\begin{align*}b\end{align*}: 3b+4<10\begin{align*}3b + 4 < 10\end{align*}. First, notice that there are two terms on the left side of the inequality, 3b\begin{align*}3b\end{align*} and 4. Therefore, use inverse operations to get the term that includes a variable, 3b\begin{align*}3b\end{align*}, by itself on one side of the inequality. In the inequality, 4 is added to 3b\begin{align*}3b\end{align*}. So, you can use the inverse of addition—subtraction. Therefore subtract 4 from both sides of the inequality. 3b+43b+443b+03b<<<<1010466 Next, use inverse operations to get the ‘b\begin{align*}b\end{align*} by itself. Since ‘3b\begin{align*}3b\end{align*} means 3×b\begin{align*}3 \times b\end{align*}, divide both sides of the inequality by 3 to solve for the variable 3b3b3b<<<6632 The answer is b<2\begin{align*}b < 2\end{align*}. ### Guided Practice Solve for n\begin{align*}n\end{align*}: 7n8n3>23\begin{align*}7n− 8n− 3 > 23\end{align*}. First, subtract 7n8n\begin{align*}7n− 8n\end{align*} because 7n\begin{align*}7n\end{align*} and 8n\begin{align*}8n\end{align*} are like terms. 7n8n37n+(8n)3n3><<232323 Next, isolate the term with the variable, n\begin{align*}-n\end{align*}, on one side of the inequality. Since 3 is subtracted from n\begin{align*}-n\end{align*}, you should add 3 to both sides of the inequality. n3n3+3n<<<2323+326 Then, since n\begin{align*}-n\end{align*} means 1n\begin{align*}-1n\end{align*} or 1×n\begin{align*}-1 \times n\end{align*}, you can divide each side of the inequality by -1 to get a positive n\begin{align*}n\end{align*} by itself on one side of the equation. Since that involves dividing both sides of the inequality by a negative number, you must reverse the inequality symbol. nn1n<<>2626126 The answer is n>26\begin{align*}n >-26\end{align*}. ### Examples #### Example 1 Solve for x\begin{align*}x\end{align*}: 4x+5<21\begin{align*}4x + 5 < 21\end{align*} First, isolate the term with the variable, 4x\begin{align*}4x\end{align*}, on one side of the inequality. Since 5 is added to 4x\begin{align*}4x\end{align*}, you should subtract 5 from both sides of the inequality. 4x+54x+554x<<<2121516 Next, since 4x\begin{align*}4x\end{align*} means 4×x\begin{align*}4 \times x\end{align*}, you can divide each side of the inequality by 4 to get ‘x\begin{align*}x\end{align*}’ by itself on one side of the inequality. 4x4x4x<<<161644 The answer is x<4\begin{align*}x < 4\end{align*}. #### Example 2 Solve for x\begin{align*}x\end{align*}: 3x6>30\begin{align*}3x-6>30\end{align*} First, isolate the term with the variable, 3x\begin{align*}3x\end{align*}, on one side of the inequality. Since 6 is subtracted from 3x\begin{align*}3x\end{align*}, you should add 6 to both sides of the inequality. 3x63x6+63x>>>3030+636 Next, since 3x\begin{align*}3x\end{align*} means 3×x\begin{align*}3 \times x\end{align*}, you can divide each side of the inequality by 3 to get ‘x\begin{align*}x\end{align*}’ by itself on one side of the inequality. 3x3x3x>>>3636312 The answer is x>12\begin{align*}x>12\end{align*}. #### Example 3 Solve for a\begin{align*}a\end{align*} 3a+2<14\begin{align*}-3a+2<14\end{align*} First, isolate the term with the variable, 3a\begin{align*}-3a\end{align*}, on one side of the inequality. Since 2 is added to 3a\begin{align*}-3a\end{align*}, you should subtract 2 from both sides of the inequality. 3a+23a+223a<<<1414212 Next, since3a\begin{align*}-3a\end{align*} means 3×a\begin{align*}-3 × a\end{align*}, you can divide each side of the inequality by -3 to get ‘a\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. 3a3a3a<<>121234 The answer is a>4\begin{align*}a>-4\end{align*}. ### Follow Up Remember the boxes of cards being sold? First, write an inequality using the given information. You want to raise at least500 by selling boxes of cards worth $5 each. You also need to pay the supplier$50 for shipping.  Let ‘b\begin{align*}b\end{align*}’ represent the number of boxes of cards.

5b50500\begin{align*}5b-50 \ge 500\end{align*}

Next, isolate the term with the variable, 5b\begin{align*}5b\end{align*}, on one side of the inequality. Since 50 is subtracted from 5b\begin{align*}5b\end{align*}, you should add 50 to both sides of the inequality.

5b505b50+505b500500+50550

Next, since 5b\begin{align*}5b\end{align*} means 5×b\begin{align*}5 \times b\end{align*}, you can divide each side of the inequality by 5 to get ‘b\begin{align*}b\end{align*}’ by itself on one side of the inequality.

5b5b5b5505505110

The answer is b110\begin{align*}b \ge 110\end{align*}.

You need to sell at least 110 boxes of cards.

### Explore More

Solve each inequality.

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

2. 4x2<10\begin{align*}4x-2<10\end{align*}

3.6y+9>69\begin{align*}6y+9>69\end{align*}

4. 2x34\begin{align*}2x-3 \le -4\end{align*}

5. 5x+28\begin{align*}5x+2 \ge -8\end{align*}

6. 2x95\begin{align*}2x-9 \le -5\end{align*}

7. x3+1>5\begin{align*}\frac{x}{3}+1>5\end{align*}

8. x21<3\begin{align*}\frac{x}{2}-1<3\end{align*}

9. x5+3>9\begin{align*}\frac{x}{5}+3>-9\end{align*}

10. x25>10\begin{align*}\frac{x}{2}-5>-10\end{align*}

11. 6k3>15\begin{align*}6k-3>15\end{align*}

12. 11+x412\begin{align*}11+\frac{x}{4}\le 12\end{align*}

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

14. 12b3b+531\begin{align*}12b-3b+5 \ge -31\end{align*}

15. 18+7n+3+6n86\begin{align*}18+7n+3+6n \le 86\end{align*}

16. 3z15z30>54\begin{align*}3z-15z-30>54\end{align*}

### Vocabulary Language: English

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 $<$, $>$, $\le$, $\ge$ and $\ne$.
Inverse Operation

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

like terms

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

## Date Created:

Dec 19, 2012

Aug 10, 2015
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