Expressions and Linear Equations · Weeks 10-18

Modeling with Inequalities

Writing and graphing inequalities to represent constraints in real world situations.

About This Topic

Writing and graphing inequalities to model real-world constraints is a key application under CCSS 7.EE.B.4b. Students represent situations where a variable must be greater than, less than, or at most a certain value, using inequality symbols and number line graphs. Unlike equations, inequalities have infinitely many solutions, and representing that solution set visually is a new and important skill.

The most conceptually challenging aspect of this topic is the rule for flipping the inequality sign when multiplying or dividing both sides by a negative number. This rule has no direct analogue in equation solving, and students who memorize it without understanding it tend to forget or misapply it. Concrete examples comparing pairs of inequalities on a number line make the logic of the flip visible.

Active learning is particularly effective here because real-world contexts give the inequality sign genuine meaning. A constraint like 'you must spend at most $40' is a clear, relatable reason to use less-than-or-equal-to. When students generate their own constraint scenarios and graph the solutions, they connect the abstract symbol to a meaningful situation.

Learning Objectives

• Formulate inequalities to represent real-world constraints involving quantities that have a minimum or maximum value.
• Graph the solution set of an inequality on a number line, accurately representing infinite solutions.
• Explain the reasoning behind flipping the inequality sign when multiplying or dividing by a negative number using concrete examples.
• Compare and contrast the solution sets of equations and inequalities in real-world contexts.
• Evaluate the reasonableness of an inequality's solution set given a specific real-world scenario.

Before You Start

Solving Two-Step Equations

Why: Students need to be proficient in isolating a variable using inverse operations before they can apply these skills to inequalities.

Representing Numbers on a Number Line

Why: Understanding how to plot points and intervals on a number line is essential for graphing the solution sets of inequalities.

Integer Operations

Why: Solving inequalities often involves multiplying or dividing by negative numbers, requiring a solid grasp of integer arithmetic.

Key Vocabulary

Inequality	A mathematical statement that compares two expressions using symbols like <, >, ≤, or ≥, indicating that one expression is not equal to the other.
Solution Set	The collection of all values that make an inequality true. This set is often infinite and represented on a number line.
Constraint	A condition or limitation that restricts the possible values of a variable in a real-world situation.
Open Circle	A notation used on a number line graph to indicate that a specific endpoint is not included in the solution set of an inequality.
Closed Circle	A notation used on a number line graph to indicate that a specific endpoint is included in the solution set of an inequality.

Active Learning Ideas

Real-World Constraint Sort

Give small groups a set of scenario cards describing real-world constraints (budget limits, minimum age requirements, temperature ranges) and a set of inequality cards. Groups match each scenario to the correct inequality and inequality symbol, then graph the solution on a number line and write one sentence explaining what the graph means in context.

⏱ 25 min·Small Groups

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Think-Pair-Share: Why Does the Sign Flip?

Write the true statement 6 > 4 on the board. Ask students to multiply both sides by -1 and determine whether the inequality sign should change. Pairs compare results and reasoning using number line checks, then share explanations with the class. Build a class explanation of the sign-flip rule from their observations.

⏱ 20 min·Pairs

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Gallery Walk: Graphing Inequalities

Post six inequalities around the room, each with a student-drawn number line graph. Three graphs are correct and three contain errors (wrong direction of arrow, incorrect open or closed circle). Pairs rotate, identify errors, and write a sticky-note correction explaining what is wrong and what the correct graph should look like.

⏱ 25 min·Pairs

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Design-a-Constraint: Create Your Own Scenario

Each student writes a real-world scenario that naturally models an inequality, writes the corresponding inequality, graphs it on a number line, and describes two specific values that satisfy the inequality and one that does not. Students share scenarios in small groups and verify each other's inequalities and graphs.

⏱ 30 min·Individual

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Real-World Connections

Budgeting for a school event: Students might need to ensure the total cost of decorations and refreshments is at most $200. They would write an inequality like c ≤ 200, where c represents the total cost.

Setting speed limits: A sign indicating a maximum speed of 45 mph translates to an inequality, v ≤ 45, where v is the vehicle's speed. Drivers must stay at or below this limit.

Planning a road trip: If a car's fuel tank holds 15 gallons and the car gets 30 miles per gallon, the maximum distance the car can travel on a full tank is 450 miles. An inequality like d ≤ 450 could represent this limit, where d is the distance traveled.

Watch Out for These Misconceptions

Common Misconception

Students do not flip the inequality sign when multiplying or dividing both sides by a negative number, applying the equation-solving rule instead.

What to Teach Instead

A concrete numerical example makes the logic clear: starting from 6 > 4 and multiplying both sides by -1 gives -6 and -4, where -6 is less than -4, so the sign must flip. Number line visualizations of the before-and-after relationship help students understand why this happens rather than just memorizing the rule.

Common Misconception

Students use a closed circle for strict inequalities (< or >) and an open circle for inclusive ones (less-than-or-equal-to or greater-than-or-equal-to), reversing the convention.

What to Teach Instead

Connect the visual to the meaning: an open circle means the endpoint is not included, while a closed or filled circle means it is included. Having students test whether the endpoint value satisfies the original inequality confirms the correct circle type.

Common Misconception

Students graph only the boundary value rather than shading the entire solution set, treating an inequality like an equation.

What to Teach Instead

Emphasize that an inequality represents all values that satisfy the condition, not just one. Ask students to name three or four specific values that make the inequality true and locate them on the number line; the shaded region emerges naturally from those points.

Assessment Ideas

Exit Ticket

Present students with the scenario: 'A baker needs to make at least 50 cookies for an order.' Ask them to: 1. Write an inequality to represent the number of cookies (c). 2. Graph the solution set on a number line. 3. Explain in one sentence what the graph shows.

Quick Check

Write the inequality -2x < 10 on the board. Ask students to solve it and graph the solution. Then, ask: 'What happens if we change it to 2x < -10? How does the graph change and why?'

Discussion Prompt

Pose the question: 'Imagine you are planning a party and have a budget of $150. You want to spend less than or equal to this amount. What are some different combinations of items you could buy? How does an inequality help you manage these choices?'

Frequently Asked Questions

Why does the inequality sign flip when you multiply or divide by a negative number?

Multiplying or dividing by a negative number reverses the order of values on the number line. For example, 4 > 2, but multiplying by -1 gives -4 < -2. The sign flips because the relative positions of the two sides swap when reflected across zero. This rule applies only to multiplication and division by a negative, not to adding or subtracting a negative.

How do you graph an inequality on a number line?

Locate the boundary value on the number line. Use an open circle if the boundary is not included (strict inequality: < or >) and a closed circle if it is included (less-than-or-equal-to or greater-than-or-equal-to). Draw an arrow from the circle in the direction of the values that satisfy the inequality: to the left for less-than, to the right for greater-than.

What is the difference between an open and a closed circle on a number line?

An open circle indicates that the boundary value is not part of the solution set. A closed (filled) circle indicates that the boundary value is included. Check by substituting the boundary value into the inequality: if it makes the inequality true, use a closed circle; if it does not satisfy the inequality, use an open circle.

How does active learning help students understand inequalities?

Real-world scenario activities give inequalities a concrete meaning: a spending limit, a minimum age, a temperature range. When students match scenarios to inequalities and graph the solution, the inequality sign and the direction of the arrow carry genuine meaning rather than being abstract conventions. Generating their own constraint scenarios deepens this connection further.