2.6 Solve Compound Inequalities

Compound inequalities combine two or more conditions using "and" or "or" to describe a range of values. They show up constantly when you need to express constraints, like acceptable temperature ranges, budget limits, or weight restrictions.

Compound Inequalities

Inequality Symbols and Solution Sets

Before working with compound inequalities, make sure you're comfortable with the four inequality symbols:

• Less than ($<$) and greater than ($>$): the endpoint is not included
• Less than or equal to ($\leq$) and greater than or equal to ($\geq$): the endpoint is included

The solution set is the collection of all values that make the inequality true. You can express it three ways: in set-builder notation, in interval notation, or as a graph on a number line.

In interval notation, parentheses $(\ )$ mark endpoints that are not included, and brackets $[\ ]$ mark endpoints that are included.

Compound Inequalities with "And"

An "and" compound inequality requires both conditions to be true at the same time. You're looking for the intersection (overlap) of the two solution sets.

These often appear in the compact form $a < x < b$, which really means $x > a$ and $x < b$.

How to solve:

1. Solve each inequality separately.
2. Find the overlap where both solutions are true at once.
3. Write the solution in interval notation or graph it on a number line.

Example: Solve $2 < x \leq 6$ and $4 \leq x < 8$.

1. The first inequality gives all values between 2 (not included) and 6 (included).
2. The second gives all values between 4 (included) and 8 (not included).
3. The overlap is the region where both are satisfied: $4 \leq x \leq 6$.
4. In interval notation: $[4, 6]$.

Compound Inequalities with "Or"

An "or" compound inequality requires at least one condition to be true. You're looking for the union (combination) of the two solution sets.

How to solve:

1. Solve each inequality separately.
2. Combine both solution sets together.
3. Write the solution using the union symbol $\cup$ in interval notation, or shade both regions on a number line.

Example: Solve $x < -1$ or $x > 3$.

1. The first inequality gives all values less than $-1$.
2. The second gives all values greater than $3$.
3. Combine them: $(-\infty, -1) \cup (3, \infty)$.

"And" vs. "Or" at a Glance

Real-World Applications

Word problems involving compound inequalities follow a consistent process:

1. Identify the variables and what they represent.
2. Translate the verbal constraints into inequalities. Use "and" when all conditions must hold, "or" when satisfying any one condition is enough.
3. Solve the compound inequality.
4. Interpret the solution back in context of the problem.

Example: A manufacturer requires the total weight of a product and its packaging to be at most 2 pounds. The product itself must weigh more than 1.5 pounds. Find the allowable weight range for the packaging.

1. Let $p$ = product weight and $w$ = packaging weight.

2. The constraints are: $p + w \leq 2$ and $p > 1.5$.

3. Since $w \leq 2 - p$ and $p$ is greater than 1.5, the most the packaging can weigh occurs when $p$ is at its smallest. As $p$ approaches 1.5, $w$ approaches $2 - 1.5 = 0.5$. For any $p$ larger than 1.5, $w$ must be even smaller.

4. The packaging must weigh less than 0.5 pounds (and of course at least 0): $0 \leq w < 0.5$.

Notice the strict inequality on 0.5. Since $p$ must be strictly greater than 1.5 (not equal to it), the packaging weight can get close to 0.5 but never actually reach it.