# 6.3: Multi-Step Inequalities

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

## Learning Objectives

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

- Solve a two-step inequality.
- Solve a multi-step inequality.
- Identify the number of solutions of an inequality.
- Solve real-world problems using inequalities.

## Vocabulary

Terms introduced in this lesson:

- two-step inequality
- multi-step inequality
- multiple solutions
- no solutions
- discrete solutions

## Teaching Strategies and Tips

In solving multi-step inequalities, remind students to follow the order of operations in each step.

In general, the steps used to solve an inequality are the same as the steps used to solve an equation. The one exception is reversing the inequality sign when multiplying or dividing by a negative.

Use Example 4 to show that an inequality can have a finite and discrete solution set. Compare and contrast this with previous inequalities having an infinite solution set.

Inequalities can have various types of solutions:

- The solution set of \begin{align*}2x \ge 10\end{align*} is the
*infinite*set \begin{align*}[5, \infty)\end{align*}. - The solution set of \begin{align*}12 + x \le x + 12\end{align*} is the
*infinite*set \begin{align*}(-\infty, -\infty)\end{align*} (all real numbers). - In the next lesson, students learn that the
*infinite*set \begin{align*}[–1, 7)\end{align*} is a solution set to an inequality. - The inequality \begin{align*}12 - x \le -x + 3\end{align*} has
*no*solutions. - Inequalities that model real-world problems in which the variables represent integer quantities (usually positive), have
*discrete*solution sets. In Example 4, the solution set is \begin{align*}\left \{0, 1, 2, 3, 4 \right \}\end{align*}, a*finite, discrete*set. In Examples 5 and 6, the solution sets are*infinite, discrete*sets:

- \begin{align*}\left \{35, 36, 37,\ldots \right \}\end{align*} and \begin{align*}\left \{146, 147, 148, \ldots \right \}\end{align*}, respectively.

## Error Troubleshooting

In Example 3a and Problems 6-10 in the *Review Questions*, remind students to follow the order of operations. Clear parentheses first.

When multiplying both sides of the inequality by \begin{align*}4\end{align*} in Example 3b, remind students to multiply *both terms* on the right by \begin{align*}4\end{align*}.

In Example 6,

- Remind students that the given numbers must be in the same unit,
*dollars*(or*cents*). - Remind students to round their answers
*up*to the next highest integer. Ask them why this is necessary.

Additional Example:

*A local bicycle shop advertises bikes for as low as \begin{align*}\$235\end{align*}. Ted decides to save his lunch money to purchase one. If he puts away \begin{align*}85\;\mathrm{cents}\end{align*} daily (including weekends), in how many days will he be able to bring home a bike?*

Hint: Convert \begin{align*}85\;\mathrm{cents}\end{align*} to \begin{align*}0.85\;\mathrm{dollars}\end{align*}. Round the answer up to \begin{align*}277\end{align*} days.