# Compound Inequalities

## Big Picture

Compound inequalities refer to a group of multiple inequalities joined by either “and” or “or”. Venn diagrams are a good visual way to represent the solutions to the compound inequalities. Each circle in a Venn diagram represents the set of solutions for each individual inequality

## Key Terms

Compound Inequality: Two or more inequalities joined by and or or.

## Types of Compound Inequalities

Inequalities joined by and

• The solution must make both inequalities true
• If a number makes only one of the inequalities true, that number is not a solution for the compound inequality
• Solutions are like the intersection of two sets. A is the set of solutions for one of the inequalities, and B is the set of solutions for the other inequalities. The solution for the compound inequality is $$A \cap B$$

Inequalities joined by or

• The solution must make at least one inequality true
• Solutions are like the union of two sets. A is the set of solutions for one of the inequalities, and B is the set of solutions for the other inequalities. The solution for the compound inequality is $$A \cup B$$

# Inequality Joined by And

$$x > a$$ and $$x < b$$ (can be rewritten as $$a < x < b$$)

• $$a$$ must be less than $$b$$

Example: $$x ≥ -40$$ and $$x < 60$$ (can be rewritten as $$-40 ≤ x < 60$$)

# Inequality Joined by Or

$$x < a \text{ or } x > b$$

• a must be greater than b

Example: $$x ≤ - 1 \text{ or } x ≥ 4$$

Remember that an open circle means that point is not included and a filled circle means that point is included.

# Inequality Joined by And

Separate the inequalities and solve them separately.

• Review the Linear Inequalities study guide on how to solve inequalities.
• Combine the solutions at the end.

Example: $$3x-5 < x+9 ≤ 5x+13$$

$$3x-5 < x+9$$
$$2x < 14$$
$$x < 7$$
and
$$x+9 ≤ 5x+13$$
$$-4 ≤ 4x$$
$$-1 ≤ x$$

Answer: $$x < 7$$ and $$x ≥ -1$$ (rewritten as $$-1 ≤ x < 7$$)

# Inequality Joined by Or

Solve each inequality separately

• Review the Linear Inequalities study guide on how to solve inequalities.

Example: $$9-2x ≤ 3 \text{ or } 3x+10 ≤ 6 - x$$

$$9-2x ≤ 3$$
$$-2x ≤ -6$$
$$x ≥ 3$$
or
$$3x+10 ≤ 6 - x$$
$$4x ≤ -4$$
$$x ≤ -1$$

Answer: $$x ≥ 3$$ and $$x ≤ -1$$