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6.3: Multi-Step Inequalities

Difficulty Level: At Grade Created by: CK-12
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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.


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.

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