Introduction

Compound inequalities allow us to describe ranges of values using two inequalities at once.
They come in two main types:

• AND inequalities (both conditions must be true)
• OR inequalities (at least one condition must be true)

You should already feel comfortable solving multi-step inequalities such as
$3x - 5 \le 10$ or $-2(x + 4) > 6$.
Now we extend those skills to situations where two inequalities work together.

What Is a Compound Inequality?

A compound inequality combines two separate inequalities using:

• AND
  • The solution must satisfy both inequalities.
  • Often written in “sandwich form”: $$a < x < b$$
• OR
  • The solution must satisfy at least one inequality.
  • Often describes values extending in two opposite directions.

Examples:

• AND: $2 < x \le 7$
• OR: $x < -3$ or $x \ge 5$

Solving AND Compound Inequalities

1. The “Sandwich” Method

Some AND inequalities appear as a single combined statement: $$a < bx + c \le d$$ To solve:

• Perform the same operation on all three parts.
• Keep inequality directions the same unless multiplying/dividing by a negative.

Example: $$-4 \le 2x + 6 < 10$$

• Subtract 6 from all parts: $$-10 \le 2x < 4$$
• Divide all parts by 2: $$-5 \le x < 2$$

2. The “Two Inequalities” Method

Sometimes the inequality is written separately:

• Solve each inequality on its own.
• The final solution is the overlap of the two solution sets.

Example: $$x - 3 > 1 \text{ and } 2x + 1 \le 9$$

• First inequality: $x > 4$
• Second inequality: $2x \le 8 \Rightarrow x \le 4$

Overlap: no values satisfy both → no solution.

Solving OR Compound Inequalities

For OR inequalities:

• Solve each inequality separately.
• The final solution is the union of the two sets.

Example: $$3x - 2 < 4 \text{ OR } x + 5 \ge 12$$

• First: $3x < 6 \Rightarrow x < 2$
• Second: $x \ge 7$

Final solution: $$x < 2 \text{ OR } x \ge 7$$

Graphing Compound Inequalities

AND (Intersection)

• Shade only where both inequalities overlap.
• Produces a single continuous interval (unless no overlap).

OR (Union)

• Shade all values that satisfy either inequality.
• Often produces two separate rays.

Graphing Tips

• Use open circles for $\lt$ or $\gt$.
• Use closed circles for $\le$ or $\ge$.
• Draw arrows for unbounded intervals.

Common Mistakes to Avoid

• Forgetting to flip the inequality when multiplying/dividing by a negative.
• Mixing up AND vs OR:
  • AND → overlap
  • OR → combine everything
• Not solving both inequalities fully.
• Assuming all AND inequalities have solutions (some don’t).

Exercises

1. Solve the compound inequality: $-3 < 2x + 1 \le 7$

   Solution

   Solve $-3 < 2x + 1 \le 7$

   • Subtract 1: $-4 < 2x \le 6$
   • Divide by 2: $-2 < x \le 3$

   Solution: $(-2, 3]$

2. Solve: $x - 4 > 3$ AND $2x + 1 \le 11$

   Solution

   %$x - 4 > 3$ AND $2x + 1 \le 11$$

   • First: $x > 7$
   • Second: $2x \le 10 \Rightarrow x \le 5$

   No overlap → no solution.

3. Solve the OR inequality: $5x - 2 \ge 13$ OR $-x + 1 < -4$

   Solution

   $$5x - 2 \ge 13$ OR $-x + 1 < -4$$

   • First: $5x \ge 15 \Rightarrow x \ge 3$
   • Second: $-x < -5 \Rightarrow x > 5$

   Union: $x \ge 3$

4. Determine whether the inequality set has a solution:
   $3x + 6 < 0$ AND $x - 2 \ge 5$

   Solution

   $$3x + 6 < 0$ AND $x - 2 \ge 5$$

   • First: $3x < -6 \Rightarrow x < -2$
   • Second: $x \ge 7$

   No overlap → no solution.

5. Write the solution set of $x < -2$ OR $x \ge 6$ in interval notation.

   Solution

   $$x < -2$ OR $x \ge 6$$ Interval notation: $$(-\infty, -2) \cup [6, \infty)$$

6. Solve: $-8 \le -2x + 4 < 6$

   Solution

   $$-8 \le -2x + 4 < 6$$

   • Subtract 4: $-12 \le -2x < 2$
   • Divide by $-2$ (flip signs): $$6 \ge x > -1$$

   Rewrite in standard order: $$-1 < x \le 6$$

7. Solve: $x + 3 > 10$ OR $4 - x \le -1$

   Solution

   $$x + 3 > 10$ OR $4 - x \le -1$$

   • First: $x > 7$
   • Second: $-x \le -5 \Rightarrow x \ge 5$

   Union: $x \ge 5$