Linear Inequalities

Active learning works well for linear inequalities because students need to see how inequalities differ from equations through multiple representations. Moving, sorting, and graphing help them grasp the idea of ranges rather than single points and why the sign flips with negatives.

• Secondary 3
• Mathematics
• 4 activities
• 25 min–40 min

Learning Objectives

1. Solve compound linear inequalities involving 'and' and 'or' conditions.
2. Represent the solution set of linear inequalities on a number line using appropriate notation.
3. Analyze the effect of multiplying or dividing an inequality by a positive or negative number.
4. Formulate linear inequalities to model real-world scenarios involving constraints or ranges.
5. Justify the choice of open or closed circles at endpoints when graphing inequalities.

Carousel Brainstorm

⏱ 30 min·Pairs

Card Sort: Inequality Matching

Prepare cards with inequalities, solution steps, number line graphs, and verbal descriptions. In pairs, students match sets correctly, then create their own cards to swap with others. Discuss mismatches as a class to reinforce sign reversal.

Prepare & details

Explain why the inequality sign reverses when multiplying or dividing by a negative number.

Facilitation Tip: During Card Sort: Inequality Matching, circulate and listen for students explaining their reasoning when matching inequalities to graphs, as this reveals gaps in understanding endpoints.

Setup: Charts posted on walls with space for groups to stand

Materials: Large chart paper (one per prompt), Markers (different color per group), Timer

• Remember
• Understand
• Analyze
• Relationship Skills
• Social Awareness

Carousel Brainstorm

⏱ 40 min·Small Groups

Relay Race: Solve and Plot

Divide class into teams. Each student solves one inequality on a card, passes to next for plotting on a shared floor number line with tape and markers. First accurate team wins; review errors together.

Prepare & details

Analyze how inequalities allow us to model ranges of possibility rather than single points of truth.

Facilitation Tip: For Relay Race: Solve and Plot, pair students strategically so those who excel with algebra can support peers during the graphing phase.

Setup: Charts posted on walls with space for groups to stand

Materials: Large chart paper (one per prompt), Markers (different color per group), Timer

• Remember
• Understand
• Analyze
• Relationship Skills
• Social Awareness

Carousel Brainstorm

⏱ 35 min·Small Groups

Budget Challenge: Real-World Inequalities

Provide scenarios like 'Phone plan costs ≤ $50 with data ≥ 10GB.' Students write, solve, and graph inequalities individually, then share in small groups to compare solution sets and endpoints.

Prepare & details

Justify the significance of the endpoint in a graphical representation of an inequality.

Facilitation Tip: In Budget Challenge: Real-World Inequalities, provide calculators to avoid arithmetic errors distracting from the inequality concepts.

Setup: Charts posted on walls with space for groups to stand

Materials: Large chart paper (one per prompt), Markers (different color per group), Timer

• Remember
• Understand
• Analyze
• Relationship Skills
• Social Awareness

Carousel Brainstorm

⏱ 25 min·Pairs

Sign Flip Demo: Balance Scales

Use physical balances with weights representing variables. Students add negative weights to one side, observe tipping, and translate to inequality sign changes. Pairs record observations and test algebraic equivalents.

Prepare & details

Explain why the inequality sign reverses when multiplying or dividing by a negative number.

Facilitation Tip: During Sign Flip Demo: Balance Scales, ask students to physically place weights on both sides to see how multiplying by a negative changes balance directions.

Setup: Charts posted on walls with space for groups to stand

Materials: Large chart paper (one per prompt), Markers (different color per group), Timer

• Remember
• Understand
• Analyze
• Relationship Skills
• Social Awareness

Teaching This Topic

Start with concrete examples before abstract rules. Use balance scales and number lines to visualize why solutions are intervals and why sign flips occur. Encourage students to explain their steps aloud, as verbalizing the logic helps internalize the process. Avoid rushing to shortcuts; ensure students understand the underlying concepts before practicing procedures.

What to Expect

Students will confidently solve inequalities, plot solutions correctly, and explain their reasoning, especially when dealing with sign changes. They will also recognize inequalities as tools for modeling real-world situations with multiple possible solutions.

Watch Out for These Misconceptions

Common Misconception

During Card Sort: Inequality Matching, watch for students who treat inequalities like equations and mark single points as solutions.

What to Teach Instead

Have them test boundary values in the inequalities to see if the endpoints satisfy the condition, using the number line from the activity to visualize inclusion or exclusion.

Common Misconception

During Relay Race: Solve and Plot, watch for students who assume inequalities always have one solution like equations do.

What to Teach Instead

Pause the race and ask groups to compare their graphs, highlighting that solutions form continuous intervals rather than isolated points.

Common Misconception

During Sign Flip Demo: Balance Scales, watch for students who ignore the sign change when dividing by negatives.

What to Teach Instead

Ask them to multiply both sides of their inequality by -1 on the balance scale, observe the shift, and explain why the inequality symbol must reverse to maintain balance.

Assessment Ideas

Quick Check

After Card Sort: Inequality Matching, ask students to solve a new inequality like 4x - 7 ≤ 13, graph it, and write the solution in interval notation. Collect their work to check for correct endpoints and graphing conventions.

Discussion Prompt

After Budget Challenge: Real-World Inequalities, ask students to share their compound inequalities and explain why an equation would not capture the budget range effectively. Listen for reasoning that connects the inequality to real-world flexibility.

Exit Ticket

During Sign Flip Demo: Balance Scales, give students two inequalities to solve: -2x > 8 and x/4 ≤ -2. Ask them to explain the difference in sign changes (or lack thereof) and graph both solutions on separate number lines before leaving class.

Extensions & Scaffolding

• Challenge students who finish early to create their own real-world inequality scenario (e.g., temperature ranges, distance limits) and trade with a partner to solve and graph.
• For students who struggle, provide pre-sorted inequality cards with either the inequality or the graph missing, so they focus on matching one element at a time.
• Deeper exploration: Have students research how inequalities are used in fields like engineering or economics, then present one example to the class with a full solution.

Key Vocabulary

Linear Inequality	A mathematical statement that compares two linear expressions using symbols like <, >, ≤, or ≥. It represents a range of values rather than a single value.
Solution Set	The collection of all values that make an inequality true. This is often represented as an interval on a number line.
Compound Inequality	An inequality that combines two or more inequalities, typically connected by 'and' or 'or', defining a more complex range of values.
Number Line Representation	A graphical method of showing the solution set of an inequality, using points, arrows, and open or closed circles to indicate the range of possible values.