English:Solving and Graphing Inequalities
Solving and Graphing Inequalities
Introduction
An inequality compares two quantities that are not necessarily equal. Instead of saying that two expressions have exactly the same value, an inequality tells you that one value is greater than, less than, greater than or equal to, or less than or equal to another value. Inequalities appear whenever there is a limit, a minimum, a maximum, a range, or more than one possible answer.
In Grades 7–8, you can use inequalities to describe real situations such as a spending limit, a minimum height requirement, a temperature range, or the number of hours available for an activity. You can also solve an inequality to find every value that makes it true and then show the full solution set on a number line.
In this aiMOOC, you will connect symbolic reasoning, number-line graphs, word problems, checking methods, and an introduction to inequalities in the coordinate plane. The main focus is on linear inequalities in one variable.
Learning Goals
By the end of this aiMOOC, you should be able to:
- Inequality symbols: Explain the meanings of <, >, ≤, and ≥ and connect them to words such as less than, more than, at most, and at least.
- Solution set: Decide whether a given number is a solution to an inequality and explain your reasoning.
- Number line: Graph one-variable inequalities using open or closed boundary points and the correct direction.
- Solving inequalities: Solve one-step and multi-step linear inequalities by using inverse operations.
- Negative numbers: Explain why multiplying or dividing both sides of an inequality by a negative number reverses the inequality sign.
- Word problems: Translate real situations into inequalities, solve them, and interpret the answer in context.
- Coordinate plane: Recognize how a two-variable linear inequality can represent a region of possible solutions.
Understanding Inequalities
Inequality Symbols and Language
The four most common inequality symbols are shown below.
| Symbol | Meaning | Example | Read as |
|---|---|---|---|
| < | Less than | x < 5 | x is less than 5 |
| > | Greater than | x > 5 | x is greater than 5 |
| ≤ | Less than or equal to | x ≤ 5 | x is at most 5 |
| ≥ | Greater than or equal to | x ≥ 5 | x is at least 5 |
Words matter. At most means a value cannot be greater than the stated amount, so it often translates to ≤. At least means a value cannot be smaller than the stated amount, so it often translates to ≥. Phrases such as more than and fewer than usually describe strict inequalities, which do not include the boundary value.
For example, if a ride requires you to be at least 140 cm tall, the height h can be modeled by h ≥ 140. A height of exactly 140 cm is included.
Solutions and Solution Sets
An equation such as x + 3 = 8 usually has a particular value that makes it true. An inequality such as x + 3 < 8 usually has many possible values. Solving the inequality gives a solution set.
For x + 3 < 8, subtract 3 from both sides:
x + 3 < 8
x < 5
Every real number less than 5 makes the original inequality true. For example, 4 works because 4 + 3 < 8 is true. The value 5 does not work because 5 + 3 < 8 becomes 8 < 8, which is false.
A useful habit is to choose a test value from your solution set and substitute it into the original inequality. This checks both your algebra and your interpretation.
Graphing Inequalities on a Number Line
A graph shows many solutions at once. The boundary value tells you where the solution set begins or ends, and the inequality sign tells you whether the boundary is included.
| Inequality type | Boundary point | Direction |
|---|---|---|
| x < a | Open circle at a | Shade left |
| x > a | Open circle at a | Shade right |
| x ≤ a | Closed circle at a | Shade left |
| x ≥ a | Closed circle at a | Shade right |
Use an open circle for < or > because the boundary is not part of the solution. Use a closed circle for ≤ or ≥ because the boundary is included. On a standard number line, values increase to the right and decrease to the left.
The following diagram shows the compound inequality -1 < x ≤ 4. The point -1 is excluded, while 4 is included.
Reading a Graph Back into Symbols
You should be able to move in both directions: from symbols to a graph and from a graph back to symbols. If a graph has a closed circle at -2 and shading to the right, then the solution is x ≥ -2. If a graph has an open circle at 6 and shading to the left, then the solution is x < 6.
This two-way connection is important because a graph is not just a picture. It is another representation of the same mathematical solution set.
Solving One-Step Inequalities
Solving an inequality is similar to solving an equation because you use inverse operations to isolate the variable. However, there is one essential difference: multiplying or dividing both sides by a negative number reverses the inequality sign.
Addition and Subtraction
Adding or subtracting the same number on both sides keeps the inequality direction unchanged.
Example:
x + 7 > 12
x > 5
Subtract 7 from both sides. The solution is x > 5.
Another example:
x - 4 ≤ 9
x ≤ 13
Add 4 to both sides. The solution is x ≤ 13.
Multiplication and Division by Positive Numbers
Multiplying or dividing both sides by the same positive number keeps the inequality direction unchanged.
Example:
4x ≥ 20
x ≥ 5
Divide both sides by 4.
Another example:
x / 3 < 6
x < 18
Multiply both sides by 3.
Multiplication and Division by Negative Numbers
When you multiply or divide both sides by a negative number, you must reverse the inequality sign.
Example:
-2x < 10
x > -5
Dividing both sides by -2 changes < to >.
Why does the sign reverse? Consider the true statement 2 < 5. Multiply both numbers by -1 and you get -2 and -5. On the number line, -2 is to the right of -5, so -2 > -5. Multiplication by a negative number reverses order.
Solving Multi-Step Inequalities
For a multi-step inequality, simplify each side and isolate the variable. A reliable order is to use the distributive property if needed, combine like terms, move variable terms to one side, move constants to the other side, and finally divide by the variable's coefficient.
Example:
3x + 5 ≤ 20
3x ≤ 15
x ≤ 5
Example with a negative coefficient:
7 - 2x > 15
-2x > 8
x < -4
The sign reverses only in the step where both sides are divided by -2.
You can check x < -4 by choosing a test value such as x = -5. Substitution gives 7 - 2(-5) > 15, which becomes 17 > 15, a true statement.
Variables on Both Sides
Some inequalities have variable terms on both sides.
Example:
5x - 2 > 3x + 6
2x - 2 > 6
2x > 8
x > 4
Check with x = 5: the left side is 23 and the right side is 21, so 23 > 21 is true.
If the variable terms cancel, you may get a statement that is always true or always false. For example, x + 4 > x + 1 simplifies to 4 > 1, which is always true, so every real number is a solution. By contrast, x + 1 > x + 4 simplifies to 1 > 4, which is always false, so there is no solution.
Compound Inequalities
A compound inequality combines two inequality conditions.
An and statement means a value must satisfy both conditions. For example, -1 < x ≤ 4 means x is greater than -1 and at most 4. The graph forms one connected interval between the two boundary points.
An or statement means a value can satisfy either condition. For example, x ≤ -3 or x > 2 has two separate parts on the number line.
Compound inequalities are a useful extension because many real situations have both a lower and an upper limit, such as a recommended temperature range or an age range for an activity.
Modeling Real Situations
Inequalities become useful when a situation has a restriction instead of one exact outcome.
Suppose you have $35 and a ticket costs $11. You want to buy snacks that cost $4 each. If s is the number of snacks, the spending condition is:
11 + 4s ≤ 35
Subtract 11:
4s ≤ 24
Divide by 4:
s ≤ 6
Because s counts snacks, it makes sense to use whole-number solutions from 0 through 6.
A different situation might involve a minimum. Suppose a reading challenge requires at least 300 pages, and you have already read 180 pages. If p is the number of additional pages, then:
180 + p ≥ 300
p ≥ 120
You need to read at least 120 more pages.
When modeling, always ask what the variable represents, whether the boundary is included, and what kinds of values make sense in context.
Extension: Inequalities in Two Variables
A two-variable inequality such as y > 2x + 1 has infinitely many ordered pairs as solutions. Its graph is a region of the coordinate plane rather than a ray on a number line.
To graph y > 2x + 1, first graph the boundary line y = 2x + 1. Because the inequality is strict, the boundary is drawn as a dashed line. Then test a point that is not on the line, such as the origin when appropriate, to determine which side of the line satisfies the inequality.
The shaded region in the diagram represents points whose coordinates satisfy y > 2x + 1.
This coordinate-plane work is an extension of the same idea you used on a number line: a boundary separates solutions from non-solutions, and the graph shows an entire set of possible answers.
Common Mistakes and How to Check Them
One common mistake is reversing the inequality sign after adding or subtracting a negative number. The sign changes only when you multiply or divide both sides by a negative number.
Another mistake is using a closed circle for a strict inequality such as x < 4. Because 4 is not a solution, the circle must be open.
A third mistake is choosing the wrong shading direction. Before drawing an arrow, say the inequality in words. For x ≥ -3, ask where numbers greater than or equal to -3 are located. They are to the right of -3.
A fourth mistake is forgetting the context of a word problem. Algebra may produce x ≤ 6.5, but if x counts whole objects, the meaningful answers may be whole numbers only.
To check your work:
- Substitution: Choose a value that should be in your solution set and test it in the original inequality.
- Boundary value: Test the boundary when the symbol is ≤ or ≥ to confirm that equality is allowed.
- Number line: Compare your symbolic answer with the direction and boundary style of your graph.
- Reasonableness: Ask whether the result makes sense in the original situation.
Interactive Tasks
Quiz: Test Your Knowledge
What does x ≤ 8 mean? (x is less than or equal to 8) (!x is greater than 8) (!x is exactly 8 only) (!x is not equal to 8)
What is the solution to x + 5 < 12? (x < 7) (!x > 7) (!x < 17) (!x > 17)
What is the solution to 3x ≥ 18? (x ≥ 6) (!x ≤ 6) (!x ≥ 15) (!x ≤ 15)
What is the solution to -2x < 10? (x > -5) (!x < -5) (!x > 5) (!x < 5)
What is the solution to x divided by 4 ≤ 3? (x ≤ 12) (!x ≥ 12) (!x ≤ 7) (!x ≥ 7)
How should x > 2 be graphed on a number line? (open circle at 2 and shade right) (!closed circle at 2 and shade right) (!open circle at 2 and shade left) (!closed circle at 2 and shade left)
How should x ≤ -1 be graphed on a number line? (closed circle at -1 and shade left) (!open circle at -1 and shade left) (!closed circle at -1 and shade right) (!open circle at -1 and shade right)
Which value satisfies 2x + 1 ≤ 7? (3) (!4) (!5) (!6)
Which inequality matches the phrase at least 12 points? (p ≥ 12) (!p > 12) (!p ≤ 12) (!p < 12)
For y > 2x + 1, how is the boundary line drawn? (dashed) (!solid) (!vertical) (!circular)
Memory Game
| Inequality | A comparison showing that two quantities are not necessarily equal |
| SolutionSet | All values that make an inequality true |
| BoundaryPoint | The endpoint where a graphed solution begins or ends |
| OpenCircle | A mark showing that the boundary value is excluded |
| ClosedCircle | A mark showing that the boundary value is included |
| ReverseRule | The rule used when multiplying or dividing both sides by a negative number |
| TestValue | A number substituted into the original inequality to check a solution |
Drag and Drop
| Match the correct terms. | Topic |
|---|---|
| Open circle | Boundary value is excluded |
| Closed circle | Boundary value is included |
| Shade left | Values are less than the boundary |
| Shade right | Values are greater than the boundary |
| Reverse the sign | Multiply or divide both sides by a negative number |
Match each graphing or solving phrase with its correct meaning.
Crossword Puzzle
| Inequality | What kind of statement compares quantities with symbols such as less than or greater than? |
| Boundary | What do you call the endpoint that separates included or excluded values on a graph? |
| Solution | What is a value called when it makes an inequality true? |
| Reverse | What must you do to the inequality direction after multiplying or dividing by a negative number? |
| Graphing | What process shows a solution set visually on a number line? |
| Variable | What symbol usually represents an unknown quantity? |
LearningApps
Cloze Text
Open-Ended Tasks
Easy
- Inequality Symbol Cards: Create four illustrated cards for <, >, ≤, and ≥. On each card, write the symbol, its meaning in words, and one original example.
- Number Line Poster: Draw a clear number line and graph four inequalities of your choice, including at least one strict inequality and one inclusive inequality.
- Solution Check: Choose three inequalities from this course, test one value that is a solution and one value that is not a solution for each, and explain what your substitutions show.
- Real-Life Inequality Sentence: Write four everyday statements involving a limit or minimum, then translate each statement into an inequality with a defined variable.
Standard
- Budget Challenge: Design a shopping situation with a fixed cost and a spending limit, write an inequality for it, solve it, graph the mathematical solution, and explain which answers make sense in context.
- Error Analysis: Invent three incorrect solutions to inequalities, including one incorrect negative-number step, and annotate each example to show where the reasoning fails and how to repair it.
- Inequality Interview: Interview a classmate, family member, or school staff member about a real rule involving at least, at most, more than, or less than; turn the rule into an inequality and explain the boundary value.
- Tutorial Video: Produce a short video that teaches how to solve and graph one inequality, including a spoken explanation of the boundary point, shading direction, and checking step.
Advanced
- Inequality Design Challenge: Create a mini set of six problems that progress from one-step inequalities to variables on both sides, provide complete solutions, and write a short teacher note explaining the intended difficulty.
- Graphing Investigation: Experiment with several inequalities that have the same boundary value but different symbols, compare their graphs, and write a conclusion about how <, >, ≤, and ≥ affect inclusion and direction.
- Two-Variable Region Project: Create an image or digital graph of three two-variable linear inequalities, label the boundary lines as solid or dashed, and use test points to justify the shaded regions.
- Constraint-Based School Plan: Choose a real school planning problem such as time, space, cost, or capacity, gather realistic data through observation or a visit to a relevant school area, model at least two constraints with inequalities, and present a reasoned recommendation.
Learning Assessment
- Equation and Inequality Comparison: Explain how solving an equation and solving an inequality are similar and different, then illustrate your explanation with one paired example.
- Negative Factor Reasoning: A student solves -3x ≥ 12 as x ≥ -4. Diagnose the error, correct the solution, and justify the sign reversal using number-line order.
- Context Translation: Create and solve an inequality for a situation with a maximum or minimum, then explain why your chosen symbol includes or excludes the boundary.
- Graph Interpretation: Given a number-line graph made by a classmate, write the matching inequality, identify two solutions and two non-solutions, and explain how the endpoint communicates inclusion.
- Multiple Representations: Represent the same inequality in words, symbols, a number-line graph, and a short real-world story, then explain how the four forms communicate the same solution set.
- Transfer to the Coordinate Plane: Explain how the boundary-and-region idea from one-variable inequalities helps you understand a shaded two-variable linear inequality.
Evidence of Learning
Important evidence of learning includes:
- Conceptual Knowledge: You accurately explain inequality symbols, boundary values, solution sets, and the negative-number reversal rule.
- Procedural Skill: You solve one-step and multi-step inequalities and show algebraic steps that preserve equivalence.
- Graphing Skill: You produce and interpret number-line graphs with correct open or closed circles and correct shading.
- Mathematical Communication: You justify decisions in clear words and connect symbols, graphs, and contexts.
- Learning Products: Your posters, models, explanations, interview notes, graphs, or videos show correct mathematics and careful reasoning.
- Transfer Achievement: You use inequalities to model unfamiliar constraints and can extend the boundary-and-solution idea to the coordinate plane.
OERs on the Topic
For further reading, explore the English Wikipedia article on linear inequalities.
Linked Learning Areas
Solving and graphing inequalities connects number sense, algebraic reasoning, graphical representation, mathematical language, and real-world modeling. It builds on equations and signed numbers and prepares you for systems of inequalities, functions, coordinate geometry, and optimization.