Compound Inequalities With And

Compound inequalities with and

A compound inequality with and means the variable must satisfy both inequalities at the same time. The answer is the overlap between the two solution sets.

1) Solve each part

Treat each inequality separately at first. If you need to add, subtract, multiply, or divide, do the same operation on both sides of each inequality.

2) Keep the inequality direction correct

• If you add or subtract the same number, the inequality sign stays the same.
• If you multiply or divide by a negative number, reverse the inequality sign.

3) Find the overlap

Once both inequalities are solved, look for the values that make both true. On a number line, this is the shared section. In interval notation, it is the intersection of the two intervals.

4) Write the simplified answer

Express the final solution as a combined inequality, a number-line interval, or both if helpful. If there is no overlap, the solution is empty.

Check your work

Pick a value inside the proposed solution and test it in both inequalities. It should make both statements true. Also test a value outside the overlap to confirm it fails at least one part.