# 6.1: Inequalities Using Addition and Subtraction

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

## Learning Objectives

At the end of this lesson, students will be able to:

- Write and graph inequalities with one variable on a number line.
- Solve an inequality using addition.
- Solve an inequality using subtraction.

## Vocabulary

Terms introduced in this lesson:

- inequality
- interval, interval of values

## Teaching Strategies and Tips

Use Examples 1 and 2 to introduce the number line as a way to graph the solution set to an inequality.

- Point out that the solution sets in Examples 1 and 2 represent all numbers which make the statements true. The solution to a linear equation is a
*number*; the solution to an inequality, an*interval of (infinite) numbers.* - Remind students that an open circle is used for inequalities containing a \begin{align*}>\end{align*} or \begin{align*}<\end{align*} symbol and a closed circle for inequalities containing a \begin{align*}\ge\end{align*} or \begin{align*}\le\end{align*} symbol.

Additional Examples:

*Graph the following inequalities on the number line.*

a. \begin{align*}x < -6\end{align*}

b. \begin{align*}x \ge 3\end{align*}

c. \begin{align*}x > 1\end{align*}

d. \begin{align*}x \le 10\end{align*}

*Write the inequality that is represented by each graph.*

a.

b.

c.

d.

In Example 3, students learn to identify inequalities in sentences. The following chart might be useful for those students having difficulty choosing the correct symbol.

\begin{align*}& < && > && \le && \ge\\ & \text{less than} && \text{greater than} && \text{at most} && \text{at least}\\ & \text{fewer} && \text{more than} && \text{no more than} && \text{no less than}\\ & &&&& \text{less than or equal to} && \text{greater than or equal to}\end{align*}

Additional Examples:

*Write each statement as an inequality and graph it on the number line.*

- You were told not to spend any more than \begin{align*}\$20\end{align*} at the arcade.
- Fewer than \begin{align*}200\end{align*} tickets are available for sale to the musical performance.
- You must be taller than \begin{align*}40\;\mathrm{inches}\end{align*} to get on this ride.
- The FDA allows for \begin{align*}30\end{align*} or more insect fragments per \begin{align*}100\;\mathrm{grams}\end{align*} of peanut butter.

Use Examples 4 and 5 to show students how to isolate variables in inequalities using addition and subtraction.

- Point out that solving inequalities is analogous to solving equations.
- The exception occurs when multiplying or dividing by a negative number. See the lesson
*Inequalities Using Multiplication and Division.*

## Error Troubleshooting

General Tip: Suggest to students having difficulty with inequalities that the inequality opens to the larger number.

- \begin{align*}a < b\end{align*} is read as “\begin{align*}a\end{align*} is less than \begin{align*}b\end{align*}” or “\begin{align*}b\end{align*} is greater than \begin{align*}a\end{align*},” depending on the perspective.
- In a statement such as \begin{align*}2 < x\end{align*}, suggest that students take the point of view of the unknown: “\begin{align*}x\end{align*} is greater than \begin{align*}2\end{align*}” or “all real numbers greater than \begin{align*}2\end{align*}” instead of “\begin{align*}2\end{align*} is less than \begin{align*}x\end{align*}”.

In Example 5d, remind students of *mixed number form* and that \begin{align*}- \frac{3} {4} - 5 = -5 \frac{3} {4}\end{align*}.