In this final chapter, we will explore compound inequalities—inequalities that combine two conditions into one statement. These appear often in real-life situations where two rules must be true at the same time, or where either one of two options is acceptable.
A compound inequality is a mathematical sentence that joins two inequalities using the words:
“and” (both conditions must be true)
“or” (at least one condition must be true)
When you see “and,” you are looking for values that make both inequalities true at the same time.
📘 Example:
3 < x ≤ 7
This means x is greater than 3 and less than or equal to 7.
✅ The solution is all numbers between 3 and 7, not including 3 but including 7.
✏️ Graph:
Open circle at 3
Closed circle at 7
Shade between
3 ◯════● 7
How to Solve “And” Inequalities
Example:
Solve: -2 ≤ x + 3 < 5
Step 1: Break it into two parts
-2 ≤ x + 3
x + 3 < 5
Step 2: Solve both:
Subtract 3: -5 ≤ x
Subtract 3: x < 2
📘 Combine:
-5 ≤ x < 2
✅ Final Answer: All x between -5 and 2, including -5 but not 2.
When you see “or,” the solution includes values that make either inequality true.
📘 Example:
x < -2 or x ≥ 3
✅ The solution is all numbers less than -2 OR greater than or equal to 3.
✏️ Graph: Two arrows going in opposite directions
←● ●→
-2 3
How to Solve “Or” Inequalities
Example:
Solve: 2x - 1 < -5 or x + 4 ≥ 9
Solve each inequality separately:
2x - 1 < -5
→ Add 1: 2x < -4
→ Divide: x < -2
x + 4 ≥ 9
→ Subtract 4: x ≥ 5
✅ Final Answer: x < -2 or x ≥ 5
Example 1: AND situation
A safe temperature range for a reptile habitat is between 72°F and 86°F (inclusive).
Let t = temperature
✅ Inequality: 72 ≤ t ≤ 86
✅ Graph: Closed circles at 72 and 86, shade between
Example 2: OR situation
A student gets extra help if their score is below 60 or above 90.
Let s = score
✅ Inequality: s < 60 or s > 90
✅ Graph: Open circle at 60 (left), open circle at 90 (right), arrows away from each other
Solve and graph each compound inequality:
-4 < x ≤ 2
x ≤ -3 or x > 1
5 ≤ 2x + 1 < 11
x - 3 < -6 or x + 4 ≥ 10
A child can ride a ride if their height is between 40 and 54 inches, inclusive.
✅ Key Takeaways:
A compound inequality joins two inequalities with “and” or “or”.
“And” means both conditions must be true (solution is an overlap).
“Or” means at least one condition must be true (solution is split).
Solve each part like a regular inequality, and then combine.
Graph using open or closed circles and shade appropriately.
You’ve learned how to:
Interpret inequality symbols
Write and solve one-step and two-step inequalities
Graph solutions on a number line
Model real-life situations with inequalities
Solve and graph compound inequalities
You're now ready to move forward to:
Graphing inequalities on a coordinate plane
Solving inequalities with absolute value
Working with systems of inequalities