How to Solve Compound Inequalities
Solving a compound inequality involves isolating the variable on one side of the inequality and then combining the two inequalities into one solution set. There are two types of compound inequalities: And and Or.
Tutor-style math help
Solve Compound Inequalities: what to notice and how to work it
Inequalities skill
Inequalities describe a set of possible values. Solve the boundary like an equation, then decide which side of the boundary makes the statement true.
What to notice first
Watch the comparison sign from the first line to the last. Multiplying or dividing by a negative reverses the direction.
Common student mistake
Do not forget open and closed endpoints. Strict signs use open circles; signs with equals use closed circles.
Key formulas and cues
\(a<b\)
\(a\le b\)
\(\text{multiply/divide by a negative} \Rightarrow \text{reverse the sign}\)
\(|x-a|<b \Rightarrow a-b<x<a+b\)
A reliable path
- Solve the boundaryTemporarily treat the inequality like an equation.
- Choose the sideUse the sign or test a number if the direction is not obvious.
- Graph the solutionUse the correct endpoint and shade the values that work.
Worked examples
Flip the sign
Example: \(-3x>12\)
- Divide both sides by -3.
- Reverse the inequality sign.
- Simplify 12 divided by -3.
Answer: \(x<-4\)
Keep the sign
Example: \(x+5\le9\)
- Subtract 5 from both sides.
- No negative multiplication or division happened.
- Keep the sign direction.
Answer: \(x\le4\)
Try one before moving on
Try: Solve \(-2x\le10\).
Answer: \(x\ge-5\). Divide by -2 and flip the sign.
Next step: do the matching worksheet or quiz while the method is still fresh, then come back and explain the first step in your own words.
x
Solve Compound Inequalities: pop-up practice
Answer these quick questions, then use the feedback to decide which part of the lesson to review.
Choose an answer to begin.
1. Solve \(x-2>5\).
2. When must the inequality sign flip?
3. \(x\le3\) uses:
Related Topics
- How to Solve Linear Inequalities
- How to Solve Radical Inequalities
- How to Solve Rational Inequalities
Step-by-step to find out how to use intercepts
To find out how to Solve Compound Inequalities, follow the step-by-step guide below:
- Simplify each inequality separately, using the properties of inequality.
- If the inequality has a “and” or “or” connective, graph the solutions for each inequality on a number line.
- If the inequality has an “and” connective, the solution is the intersection of the two graphs. If the inequality has an “or” connective, the solution is the union of the two graphs.
- Write the solution in interval notation.
- Check your solution by plugging a test point from the solution set back into the inequality.
Two types of compound inequalities:
- And compound inequality: When solving an “and” compound inequality, the solution set is the intersection of the two inequalities. For example, to solve the inequality \(4<x+2<8\), we can first subtract \(2\) from each side to get \(2<x<6\). Then we can take the intersection of the solutions of these two inequalities to get \(x>2\) and \(x<6\). The solution set is therefore \(2<x<6\).
- Or compound inequality: When solving an “or” compound inequality, the solution set is the union of the two inequalities. For example, to solve the inequality \(4 < x + 2 < 8\) or \(x – 5 > 0\) we can first solve the first inequality \(4 < x + 2 < 8\) to get \(2 < x < 6\) and then the second inequality \(x – 5 > 0\) to get \(x > 5\). Then we can take the union of the solutions of these two inequalities to get \(x > 2\) or \(x > 5\). The solution set is therefore \(x > 2\) or \(x > 5\).
Note: It’s important to note that when solving compound inequalities, it’s important to follow the order of operations, and to be careful when using the less than or greater than sign as they can flip the inequality.
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