Inequalities: Solving & Graphing

Active learning works for inequalities because the flip rule and graphing conventions are abstract and easily confused. Hands-on tasks let students test their own reasoning, catch mistakes in real time, and connect the symbol changes to the number line they can see and move.

• 8th Grade
• Mathematics
• 4 activities
• 20 min–40 min

Learning Objectives

1. Calculate the solution set for one-variable linear inequalities.
2. Compare the process of solving linear equations with solving linear inequalities, identifying key differences.
3. Explain the effect of multiplying or dividing an inequality by a negative number on the solution set.
4. Construct a number line graph that accurately represents the solution set of a given linear inequality.
5. Analyze the impact of strict versus inclusive inequality symbols on the graphical representation of the solution set.

Think-Pair-Share

⏱ 20 min·Pairs

Think-Pair-Share: The Flip Rule

Present -2x < 6 and ask students to solve it individually. Pairs compare: did both students flip the inequality? Each pair must explain to the class why multiplying or dividing by a negative number requires flipping. The class then tests the rule by substituting specific numbers into both the original and solved inequality.

Prepare & details

Differentiate between solving equations and solving inequalities.

Facilitation Tip: During Think-Pair-Share, circulate and listen for pairs who correctly verbalize when the flip rule is needed, not just repeating the rule.

Setup: Standard classroom seating; students turn to a neighbor

Materials: Discussion prompt (projected or printed), Optional: recording sheet for pairs

• Understand
• Apply
• Analyze
• Self-Awareness
• Relationship Skills

Inquiry Circle

⏱ 25 min·Small Groups

Inquiry Circle: Test the Solution Set

Groups graph a linear inequality on a number line, then test five specific values: two from the shaded region, two from the unshaded region, and the boundary value. They substitute each into the original inequality to verify which satisfy it, then write a general statement about what the graph represents.

Prepare & details

Explain the impact of multiplying or dividing by a negative number on an inequality.

Facilitation Tip: In Collaborative Investigation, hand out pre-solved inequalities with deliberate errors so groups must trace each step and decide if the flip was applied appropriately.

Setup: Groups at tables with access to source materials

Materials: Source material collection, Inquiry cycle worksheet, Question generation protocol, Findings presentation template

• Analyze
• Evaluate
• Create
• Self-Management
• Self-Awareness

Gallery Walk

⏱ 30 min·Small Groups

Gallery Walk: Open or Closed?

Post eight inequalities with their number line graphs. Students rotate in groups, identifying any graphing errors (wrong circle type, wrong shading direction) and correcting them on sticky notes. Groups discuss their corrections during a whole-class debrief.

Prepare & details

Construct a graph that accurately represents the solution set of a linear inequality.

Facilitation Tip: At the Gallery Walk, ask students to annotate each poster with sticky notes that restate the boundary test they performed for the circle type.

Setup: Wall space or tables arranged around room perimeter

Materials: Large paper/poster boards, Markers, Sticky notes for feedback

• Understand
• Apply
• Analyze
• Create
• Relationship Skills
• Social Awareness

Stations Rotation

⏱ 40 min·Small Groups

Stations Rotation: Solve, Graph, Interpret

Four stations build the full process: (1) solve the inequality, (2) graph the solution set on a number line, (3) interpret the solution in a real-world context such as safe weight limits, (4) write your own inequality from a verbal constraint. Students rotate through all four stations.

Prepare & details

Differentiate between solving equations and solving inequalities.

Facilitation Tip: During Station Rotation, place a mini whiteboard at each station where students must show their solution, graph, and a quick reason for the circle choice before rotating on.

Setup: Tables/desks arranged in 4-6 distinct stations around room

Materials: Station instruction cards, Different materials per station, Rotation timer

• Remember
• Understand
• Apply
• Analyze
• Self-Management
• Relationship Skills

Teaching This Topic

Teach the flip by having students multiply the same inequality by 2 and then by -2 on the number line, so they see the order of points reverse. Avoid teaching tricks like “flip when you see a negative,” because students then apply the rule inappropriately. Use the phrase “multiply or divide both sides by a negative” consistently to anchor the concept.

What to Expect

Students solve inequalities correctly, explain when and why the sign flips, and graph with accurate open or closed circles. They justify their steps to peers and adjust their work based on feedback from classmates or the number line itself.

Watch Out for These Misconceptions

Common Misconception

During Think-Pair-Share: The Flip Rule, watch for students who claim the sign flips whenever a negative appears anywhere in the problem.

What to Teach Instead

Have the pair re-examine the step where both sides were multiplied or divided by a negative; ask them to mark only that operation and ignore negatives that are added or subtracted.

Common Misconception

During Gallery Walk: Open or Closed?, watch for students who assume all boundary points use closed circles.

What to Teach Instead

Direct them back to the original inequality and have them substitute the boundary value to verify inclusion before deciding on the circle type.

Assessment Ideas

Exit Ticket

After Collaborative Investigation: Test the Solution Set, give each student the inequality -4x + 1 ≥ 9 to solve and graph. Collect work to check that the flip is applied correctly and the circle is closed.

Quick Check

During Station Rotation: Solve, Graph, Interpret, show students two statements on the board: 'Solving inequalities is exactly the same as solving equations.' and 'Multiplying or dividing an inequality by a negative number does not change the inequality symbol.' Ask students to circle True or False and write one sentence explaining their choice on the back of their mini whiteboard.

Discussion Prompt

After Think-Pair-Share: The Flip Rule, ask students to explain to the class how they would use a number line and the inequality -2x < 6 to show why the symbol flips when dividing by -2. Circulate and listen for references to the order of numbers on the number line.

Extensions & Scaffolding

• Challenge students to write three inequalities that produce the same solution set but require different flips or no flip at all.
• Scaffolding: Provide partially solved inequalities with blanks for the flip step and ask students to fill in the missing operation and sign.
• Deeper exploration: Ask students to generate a real-world scenario that can be modeled by a compound inequality and solve it graphically.

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.
Solution Set	The collection of all values that make an inequality true. This is often represented graphically on a number line.
Open Circle	A symbol used on a number line graph to indicate that the boundary point is not included in the solution set (used for < and > inequalities).
Closed Circle	A symbol used on a number line graph to indicate that the boundary point is included in the solution set (used for ≤ and ≥ inequalities).
Reversal Property	The rule stating that when multiplying or dividing both sides of an inequality by a negative number, the direction of the inequality symbol must be reversed.