6.3: MultiStep Inequalities
Learning Objectives
At the end of this lesson, students will be able to:
 Solve a twostep inequality.
 Solve a multistep inequality.
 Identify the number of solutions of an inequality.
 Solve realworld problems using inequalities.
Vocabulary
Terms introduced in this lesson:
 twostep inequality
 multistep inequality
 multiple solutions
 no solutions
 discrete solutions
Teaching Strategies and Tips
In solving multistep 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
2x≥10 is the infinite set[5,∞) .  The solution set of
12+x≤x+12 is the infinite set(−∞,−∞) (all real numbers).  In the next lesson, students learn that the infinite set
[–1,7) is a solution set to an inequality.  The inequality
12−x≤−x+3 has no solutions.  Inequalities that model realworld problems in which the variables represent integer quantities (usually positive), have discrete solution sets. In Example 4, the solution set is
{0,1,2,3,4} , a finite, discrete set. In Examples 5 and 6, the solution sets are infinite, discrete sets:

{35,36,37,…} and{146,147,148,…} , respectively.
Error Troubleshooting
In Example 3a and Problems 610 in the Review Questions, remind students to follow the order of operations. Clear parentheses first.
When multiplying both sides of the inequality by
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
Hint: Convert