Mathematics · 7th Grade · Expressions and Linear Equations · Weeks 10-18

Solving Two-Step Inequalities

Students will solve and graph two-step linear inequalities.

• Common Core State Standards
• CCSS.Math.Content.7.EE.B.4b

About This Topic

Two-step linear inequalities extend the procedural and conceptual work of one-step inequalities under CCSS 7.EE.B.4b. Students apply inverse operations in sequence while tracking whether any step requires flipping the inequality sign, then represent the solution set on a number line. The combination of multi-step solving and graphing makes this one of the more complex skills in the Expressions and Equations unit.

Students encounter real-world scenarios that require two constraints to be applied in sequence: a spending limit that also has a minimum purchase requirement, or a speed limit applied after accounting for a buffer. These contexts make two-step inequalities meaningful and give students a way to verify whether their solution makes sense.

Active learning strategies that move students from solving to interpreting and back are especially valuable here. When students write their own word problems for a given two-step inequality and share them with peers, they must think carefully about what each step of the solution represents in context. This bidirectional thinking builds the algebraic fluency that carries into high school mathematics.

Key Questions

1. Explain the process for solving two-step inequalities, including when to reverse the inequality sign.
2. Analyze real-world scenarios that can be modeled and solved using two-step inequalities.
3. Construct a two-step inequality from a given verbal description.

Learning Objectives

• Solve two-step linear inequalities involving addition, subtraction, multiplication, and division, accurately reversing the inequality sign when necessary.
• Graph the solution set of two-step linear inequalities on a number line, using appropriate notation for open and closed circles and direction of shading.
• Analyze real-world scenarios to construct and solve two-step inequalities, interpreting the solution in the context of the problem.
• Explain the algebraic steps and reasoning required to solve a two-step inequality, including the rule for reversing the inequality sign.

Before You Start

Solving One-Step Inequalities

Why: Students must be proficient in isolating a variable using one inverse operation and understanding the concept of reversing the inequality sign before tackling two-step problems.

Solving Two-Step Equations

Why: Familiarity with applying inverse operations in sequence to isolate a variable is essential for the procedural aspect of solving two-step inequalities.

Key Vocabulary

Inequality	A mathematical statement that compares two expressions using symbols such as <, >, ≤, or ≥, indicating that one expression is less than, greater than, less than or equal to, or greater than or equal to the other.
Two-step inequality	An inequality that requires two inverse operations to isolate the variable, such as combining addition/subtraction with multiplication/division.
Reverse the inequality sign	To flip the inequality symbol from < to >, or > to <, which must be done when multiplying or dividing both sides of an inequality by a negative number.
Solution set	The collection of all values for the variable that make the inequality true.

Watch Out for These Misconceptions

Common Misconception

Students flip the inequality sign when dividing by a positive number, applying the rule too broadly after learning it for negative divisors.

What to Teach Instead

Clarify that the sign flips only when multiplying or dividing both sides by a negative number. Annotating the sign of the divisor before applying the rule helps students check each division step. Error analysis activities where students identify unnecessary flips are effective for correcting this overgeneralization.

Common Misconception

Students complete the first step correctly but forget to flip the sign in the second step when the second operation involves dividing by a negative number.

What to Teach Instead

Encourage step-by-step annotation where each operation is written out fully, including the sign of the multiplier or divisor, before simplifying. Having partners check specifically whether any step involved a negative divisor and whether the sign was handled correctly reduces this error.

Common Misconception

Students graph the solution set at the boundary value only rather than representing the full range of values.

What to Teach Instead

Require students to name three specific values in the solution set and locate them on the number line before drawing the arrow. The pattern of points makes the continuous nature of the solution set clear and guides correct graphing.

Active Learning Ideas

Step-by-Step Annotation: Show Every Operation

Present a two-step inequality and require students to solve it in writing with a full annotation for every step, including 'added 3 to both sides' and 'divided both sides by -2, so flip the sign.' Partners swap papers and verify that every annotation is correct and every operation was applied to both sides.

20 min·Pairs

Real-World Scenario Match

Provide small groups with sets of two-step inequality cards and scenario cards. Groups match each inequality to a real-world context, solve the inequality, and explain what the solution set means in the scenario. Groups present their most interesting match to the class and explain why it was a good fit.

30 min·Small Groups

Create-a-Problem: Two-Step Inequality to Word Problem

Each pair receives a two-step inequality, solves it, and writes a word problem that the inequality models. They swap with another pair to solve and evaluate whether the word problem accurately reflects the inequality. Authors explain any discrepancies and revise the word problem if needed.

30 min·Pairs

Error Analysis: Find the Flip

Present four solved two-step inequalities, two of which contain errors related to the sign flip (failing to flip when dividing by a negative, or flipping unnecessarily when dividing by a positive). Small groups identify which solutions are correct, locate and explain errors, and write corrected solutions with the sign flip step explicitly annotated.

25 min·Small Groups

Real-World Connections

• A student saving money for a video game might have a goal amount and a weekly allowance. They can set up a two-step inequality to determine how many weeks it will take to reach their goal after accounting for initial savings.
• A delivery driver has a maximum daily mileage limit. After driving a fixed distance to their first stop, they can use a two-step inequality to calculate the maximum distance they can travel on subsequent deliveries for the rest of the day.

Assessment Ideas

Exit Ticket

Provide students with the inequality 3x - 5 < 10. Ask them to solve for x, graph the solution on a number line, and write one sentence explaining why they did or did not need to reverse the inequality sign.

Quick Check

Present students with a word problem: 'Maria wants to buy a book that costs $15. She has $5 saved and earns $2 per hour for babysitting. Write and solve a two-step inequality to find the minimum number of hours she needs to babysit.' Review student responses for accuracy in setting up and solving the inequality.

Discussion Prompt

Pose the question: 'When solving the inequality -2y + 7 > 15, what is the first step and why? What is the second step and why? What is the final solution and how would you graph it?' Facilitate a class discussion where students explain their reasoning for each step.

Frequently Asked Questions

How do you solve a two-step inequality like 2x - 3 > 7?

Add 3 to both sides to get 2x > 10, then divide both sides by 2 to get x > 5. Because you divided by a positive number, the inequality sign does not change. Graph by placing an open circle at 5 and drawing an arrow to the right, since all values greater than 5 satisfy the inequality.

When does the inequality sign reverse in a two-step inequality?

The sign reverses in any step where you multiply or divide both sides by a negative number. In a two-step inequality, this can happen at either step. If both steps involve dividing by positive numbers, the sign never changes. Check the sign of each multiplier or divisor as you work through each step.

How do you write a two-step inequality from a word problem?

Identify the unknown and assign it a variable. Translate the rate or multiplier into a coefficient and the additional amount into a constant. Identify the constraint word: 'at most' means less-than-or-equal-to, 'at least' means greater-than-or-equal-to, 'more than' means >, 'fewer than' means <. Assemble the inequality, solve, and check that the solution set makes sense in the context.

How does active learning support students solving two-step inequalities?

When students write word problems for a given two-step inequality and share them with peers, they must interpret what each algebraic step means in context. This forces precision in understanding both the inequality structure and the solution set. Error analysis activities focused on the sign-flip rule help students distinguish when it applies from when it does not, which is a common point of confusion.