Solving and Graphing Inequalities

Learn how to solve one-step and two-step inequalities and represent solutions on a number line.

Intermediate25 minLesson
Printable worksheet

Definition

An inequality is a mathematical statement that compares two expressions using inequality symbols:
SymbolMeaningExample
<<Less thanx<5x < 5
>>Greater thanx>3x > 3
≤\leqLess than or equal tox≤7x \leq 7
≥\geqGreater than or equal tox≥−2x \geq -2
Unlike equations (which have one solution), inequalities have infinitely many solutions.
For example, x>3x > 3 means xx can be 4, 5, 10, 3.5, or any number greater than 3.
Key Rule: When you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign.
−2x>6  ⟹  x<−3-2x > 6 \implies x < -3

Try it now

Which symbol means 'greater than or equal to'?

Worked Examples

Solve x+5>12x + 5 > 12 and graph the solution.
1

Identify the operation

5 is being added to xx → Need to subtract 5
2

Subtract 5 from both sides

x+5−5>12−5x + 5 - 5 > 12 - 5 → x>7x > 7
3

Graph on number line

Open circle at 7, shade to the right → All numbers greater than 7

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: −2x>8-2x > 8 becomes x<−4x < -4 (not x>−4x > -4). The sign MUST flip.

Using closed circle for strict inequalities (<< or >>)

Why it's wrong: Strict inequalities (<<, >>) do NOT include the boundary value.

Correct: x>5x > 5: Open circle at 5 (5 is not included). x≥5x \geq 5: 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 x>3x > 3, test x=5x = 5. Since 5>35 > 3 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: x−(−5)>2x - (-5) > 2 becomes x+5>2x + 5 > 2, then x>−3x > -3. No flip needed for subtraction.

Interactive Visual

to

x ≥ 4·[4, ∞)

Click on numbers to select them. Adjust the range to explore different values.

to

x < -5·(−∞, -5)

Click on numbers to select them. Adjust the range to explore different values.

Interactive Sandbox

Expression Calculator

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Practice Problems

15 problems
Problem 1 of 15
Easy
Which symbol means 'greater than or equal to'?

Why It Matters

Inequalities describe real-world situations with ranges and limits:
  • Age requirements: You must be at least 16 to drive (age≥16\text{age} \geq 16)
  • Speed limits: Drive no faster than 50 km/h (speed≤50\text{speed} \leq 50)
  • Budget constraints: Spend less than 100 dollars (cost<100\text{cost} < 100)
  • Temperature ranges: Water stays liquid between 0 and 100 degrees Celsius
  • Grades: You need at least 60 points to pass (score≥60\text{score} \geq 60)
Inequalities help us express conditions, constraints, and ranges that appear everywhere in daily life!

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? 25t+15≤9025t + 15 \leq 90

1Try It Yourself

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

Solve 12m<15012m < 150. What is the largest whole number of months?

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? 4s≥2004s \geq 200, so s≥50s \geq 50 km/h.

2Try It Yourself

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 speed=distancetime\text{speed} = \frac{\text{distance}}{\text{time}}, then 300t≤80\frac{300}{t} \leq 80. Solve for the smallest tt (in hours).

Key Takeaways

  • 1Inequalities compare expressions using <<, >>, ≤\leq, ≥\geq
  • 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 ≤\leq and ≥\geq
  • 5Shade the number line in the direction of the solution set

Frequently Asked Questions

Only when multiplying or dividing both sides by a negative number. Adding or subtracting (even negative numbers) does not require flipping.
Only when multiplying or dividing both sides by a negative number. Adding or subtracting (even negative numbers) does not require flipping.
<< means strictly less than (not equal), shown with an open circle. ≤\leq means less than OR equal to, shown with a closed circle.
Pick a value from your solution set and substitute it back into the original inequality. If it makes the inequality true, your answer is likely correct.

Glossary

Inequality
A mathematical statement comparing two expressions using <<, >>, ≤\leq, or ≥\geq
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 (≤\leq or ≥\geq)
Boundary value
The value where the inequality changes from true to false
Today's challengeOne problem, three tries

More in This Topic

Introduction to Inequalities

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Graphing Inequalities on a Number Line

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One-Step Inequalities

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Two-Step Inequalities

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