Solving and Graphing Inequalities
Learn how to solve one-step and two-step inequalities and represent solutions on a number line.
Definition
| Symbol | Meaning | Example |
|---|---|---|
| Less than | ||
| Greater than | ||
| Less than or equal to | ||
| Greater than or equal to |
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Worked Examples
Solve and graph the solution.
Identify the operation
5 is being added to → Need to subtract 5
Subtract 5 from both sides
→
Graph on number line
Open circle at 7, shade to the right → All numbers greater than 7
Answer: (open circle at 7, arrow pointing right)
Common Mistakes
Forgetting to flip the sign when dividing by a negative
Why it's wrong: The most common error! When dividing (or multiplying) by a negative number, the inequality direction reverses.
Correct: becomes (not ). The sign MUST flip.
Using closed circle for strict inequalities ( or )
Why it's wrong: Strict inequalities (, ) do NOT include the boundary value.
Correct: : Open circle at 5 (5 is not included). : Closed circle at 5 (5 is included).
Shading in the wrong direction
Why it's wrong: Students sometimes shade based on the sign without checking the variable's position.
Correct: Always test a value: For , test . Since is true, shade the region containing 5.
Flipping the sign when subtracting negatives
Why it's wrong: The flip rule ONLY applies to multiplication and division by negatives, not addition/subtraction.
Correct: becomes , then . No flip needed for subtraction.
Interactive Visual
Click on numbers to select them. Adjust the range to explore different values.
Balance Scale
Solution: x = 4
Interactive Sandbox
Expression Calculator
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History
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Practice Problems
15 problemsWhich symbol means 'greater than or equal to'?
Why It Matters
- Age requirements: You must be at least 16 to drive ()
- Speed limits: Drive no faster than 50 km/h ()
- Budget constraints: Spend less than 100 dollars ()
- Temperature ranges: Water stays liquid between 0 and 100 degrees Celsius
- Grades: You need at least 60 points to pass ()
Real World Applications
Budget Planning
Determine how many items you can buy while staying within a budget.
Example:
Concert tickets cost 25 dollars each, plus a 15 dollar service fee. If you have 90 dollars, how many tickets can you buy?
A streaming service costs 12 dollars per month. You want to spend less than 150 dollars per year on streaming.
For how many months can you subscribe?
Step 1: Write the mathematical expression
Write the inequality:
Speed and Distance
Calculate safe driving speeds to arrive on time.
Example:
You need to drive at least 200 km in 4 hours. What average speed is required? , so km/h.
A delivery truck must travel 300 km. To avoid a fine, the driver must average no more than 80 km/h.
What is the minimum time for the trip?
Step 1: Write the mathematical expression
If , then
Key Takeaways
- 1Inequalities compare expressions using , , ,
- 2Solve inequalities like equations: add, subtract, multiply, divide on both sides
- 3Critical rule: Flip the inequality sign when multiplying or dividing by a negative number
- 4Graph solutions: open circle for and , closed circle for and
- 5Shade the number line in the direction of the solution set
Frequently Asked Questions
Glossary
- Inequality
- A mathematical statement comparing two expressions using , , , or
- Solution set
- All values that make an inequality true
- Open circle
- Used on a number line to show a value is NOT included ( or )
- Closed circle
- Used on a number line to show a value IS included ( or )
- Boundary value
- The value where the inequality changes from true to false