Compound Inequalities

TipLearning Objectives

By the end of this lesson, you’ll be able to:

  • Interpret and solve compound inequalities using and or or.
  • Write and understand interval notation.
  • Represent solution sets on a number line.

Key Ideas

A compound inequality joins two inequalities using and or or.

Number lines illustrating compound inequalities: an AND interval versus OR split intervals.

AND (both conditions must be true)

\[ a < x < b \]
Means \(x\) is between \(a\) and \(b\).

OR (either condition may be true)

\[ x < -2 \quad \text{or} \quad x > 5 \]
Means two separate intervals.

Common Problem Types

1. AND Inequalities

Chained style: \(a < bx + c < d\)

2. OR Inequalities

Separate inequalities joined by “or.”

3. Solving in Steps

Solve each inequality individually, then combine the results.

4. Interpreting Graphs

Recognize when intervals overlap (AND) or split (OR).

Worked Examples

Example 1 — AND

\[ -1 < 2x + 3 < 7 \]

Subtract 3 → \(-4 < 2x < 4\)
Divide → \(-2 < x < 2\)

Example 2 — OR

Solve: \[ 3x - 1 \ge 5 \quad \text{or} \quad 2x + 4 < 0 \]

Left: \(x \ge 2\)
Right: \(x < -2\)

Final: \(x < -2\) or \(x \ge 2\)

WarningCommon Mistakes
  • Mixing up AND and OR shapes.
  • Forgetting to flip the inequality when dividing by a negative.
  • Combining the solutions incorrectly.

Practice Problems

  1. \(1 < 2x + 5 < 9\)
  2. \(x - 4 \le -3\) or \(2x + 1 \ge 7\)
  3. \(-6 \le 3x + 6 \le 9\)
  4. \(2x - 3 > 5\) or \(x + 1 < 0\)

1. \(-2 < x < 2\)
2. \(x \le 1\) or \(x \ge 3\)
3. \(-4 \le x \le 1\)
4. \(x > 4\) or \(x < -1\)

Summary

  • AND → overlap region; OR → two separate intervals.
  • Solve each inequality separately, then combine logically.
  • Flip sign only when dividing by a negative.
  • “AND” = intersection. “OR” = union.
  • Use number lines to visualize split vs overlap.
  • Check each solution interval separately.