Compound Inequalities- Solving Multiple Conditions

What Are Compound Inequalities?

A compound inequality is just two or more inequalities joined together. Instead of solving one condition, you're solving multiple conditions at the same time. That's it. That's the whole concept.

You see them in forms like:

-3 < x + 2 ≤ 5

or

x < -1 OR x > 3

The tricky part isn't the math—it's understanding whether you're dealing with an AND situation or an OR situation. Get that wrong, and everything falls apart.

The Two Types: AND vs OR

This is where most people mess up. The connector word matters more than you think.

AND Inequalities

With AND, both conditions must be true simultaneously. The solution is the overlap (intersection) of both conditions. Your answer has to satisfy every inequality in the compound.

Example: -2 < x ≤ 4

This means x is greater than -2 AND x is less than or equal to 4. So x lives in that narrow band between -2 and 4.

OR Inequalities

With OR, only one condition needs to be true. The solution is everything that satisfies at least one of the inequalities. As long as one condition works, you're good.

Example: x < -1 OR x > 3

This means x can be less than -1, OR x can be greater than 3. Anything in between (-1 to 3) doesn't work.

How to Solve AND Compound Inequalities

Solving AND inequalities is straightforward. You work with the whole thing at once.

Example: Solve -4 < 2x - 2 < 6

Step 1: You have three parts. Keep the inequality balanced by doing the same thing to all three.

Step 2: Add 2 to all parts.

-2 < 2x < 8

Step 3: Divide all parts by 2.

-1 < x < 4

Your solution: x is between -1 and 4, not including -1 but including 4.

⚠️ Critical rule: When you multiply or divide by a negative number, flip ALL inequality signs. Forgetting this is the #1 mistake students make.

How to Solve OR Compound Inequalities

OR inequalities are solved separately, then combined.

Example: Solve 3x + 1 < 4 OR 2x - 5 > 3

Step 1: Solve each inequality independently.

First inequality:

3x + 1 < 4

3x < 3

x < 1

Second inequality:

2x - 5 > 3

2x > 8

x > 4

Step 2: Combine with OR.

x < 1 OR x > 4

That's your solution. Anything less than 1 works. Anything greater than 4 works. The middle (1 to 4) is excluded.

Graphing Compound Inequalities

Graphing helps you visualize the solution. Here's how to do it on a number line.

For AND inequalities (overlap):

• Draw the first condition on the number line
• Draw the second condition on the same line
• The solution is where both lines overlap
• Use closed circles for ≤ or ≥
• Use open circles for < or >

For OR inequalities (union):

• Draw both conditions on the same number line
• The solution is everything covered by either line
• Closed or open circles depend on the inequality symbols

📍 Quick tip: A closed circle (●) means the point is included. An open circle (○) means it's not included.

Solving Inequalities with Absolute Values

Absolute value compound inequalities show up constantly. They actually represent an AND situation disguised as a single expression.

|x - 3| < 5 becomes -5 < x - 3 < 5

Then solve normally:

-5 < x - 3 < 5

-5 + 3 < x < 5 + 3

-2 < x < 8

For |expression| > value, you get an OR situation:

|x + 1| > 2 becomes x + 1 < -2 OR x + 1 > 2

Which simplifies to x < -3 OR x > 1

Quick Reference Table

Common Mistakes to Avoid

• Forgetting to flip the sign when multiplying or dividing by negatives. Every. Single. Time.
• Confusing AND with OR. Draw a number line if you're unsure.
• Treating OR like AND. With OR, you don't need both conditions to work—just one.
• Losing track of open vs closed circles on the graph. Check your inequality symbols.
• Not solving absolute values correctly. Remember: < means AND, > means OR.

Practice: Solve This

Try solving -1 ≤ 3 - 2x < 7

Break it down:

You have -1 ≤ 3 - 2x AND 3 - 2x < 7

Subtract 3 from all parts:

-4 ≤ -2x < 4

Divide by -2 (remember to flip the signs):

2 ≥ x > -2

Rewrite as -2 < x ≤ 2

That's your answer. x is between -2 and 2, excluding -2 but including 2.

Bottom Line

Compound inequalities aren't complicated. You have two skills: solving basic inequalities and understanding AND vs OR. Get those down and you can solve anything in this topic.

Most errors come from rushing. Slow down, write out each step, and check your inequality direction when you multiply or divide by negatives. That's literally all there is to it.