Lesson 82: Solving Multi-Step Compound Inequalities

New Concept

A compound inequality is made of two inequalities joined by the word AND or OR. For example, $-2 < x \operatorname{AND} x \le 5$.

What’s next

Next, you’ll apply familiar algebraic properties to isolate the variable and find the solution set for these more complex statements.

Property

An 'AND' compound inequality, often written in a compact form like a < x < b, is true only if a value for x satisfies both conditions at the same time.

Explanation

Think of your variable as being trapped between two walls! To solve for it, you have to do the same thing to all three parts of the inequality—the left side, the middle, and the right side. It’s a package deal; whatever inverse operation you perform, apply it everywhere to isolate the variable.

Examples

To solve $-5 \le 2x + 1 \le 9$, subtract 1 from all three parts to get $-6 \le 2x \le 8$. Then, divide all parts by 2 to find that $-3 \le x \le 4$.
To solve $10 < 2(x + 3) < 24$, first distribute to get $10 < 2x + 6 < 24$. Then subtract 6 from all parts: $4 < 2x < 18$. Finally, divide by 2: $2 < x < 9$.

Let's untangle this compact inequality by applying our rules to all three parts at once.

Example Problem

Solve the inequality $-21 \le 3(2x - 5) \le 33$ and justify each step.

Step-by-Step

1. Start with the given inequality.

2. First, use the Distributive Property on the middle part of the inequality.

3. Now, apply the Addition Property of Inequality by adding $15$ to all three parts to start isolating the variable term.

4. Simplify each part of the inequality.

5. Use the Division Property of Inequality by dividing all three parts by $6$ to solve for $x$.

6. Simplify to find the final solution.

Property

An 'OR' compound inequality is true if a value satisfies at least one of the two separate inequalities.

Explanation

'OR' gives you choices! The solution can be in one zone or another, so you solve each inequality completely on its own, like they're two different problems. Your final answer is the combination of all possible solutions from both parts, and the graph often shows arrows pointing away from each other.

Examples

Solve $4x - 5 < -13$ OR $3x + 2 > 14$. The first part gives $4x < -8$, so $x < -2$. The second gives $3x > 12$, so $x > 4$. The solution is $x < -2$ OR $x > 4$.
Solve $-10 > 2(x - 1)$ OR $15 < 3(x + 1)$. First part: $-5 > x - 1$, so $-4 > x$. Second part: $5 < x + 1$, so $4 < x$. The solution is $x < -4$ OR $x > 4$.

Property

Use the formula $\text{Min Value} \leq \frac{\text{Sum of Data}}{\text{Number of Data}} \leq \text{Max Value}$ to find a missing value in an average.

Explanation

Need to figure out what score you need on a final test? This is how! You trap the average formula between your minimum and maximum goals. Then, you just use inverse operations to work backward and find the range of scores you need to land on target. It’s real-world algebra for hitting your goals!

Examples

Felipe's test scores are 94, 88, and 91. His average must be between 90 and 100. Find the fourth score, x: $90 \le \frac{94+88+91+x}{4} \le 100$. This becomes $360 \le 273+x \le 400$, so he must score $87 \le x \le 127$.
Four babies have an average weight between 6 and 8 pounds. Three weigh 5.2, 6.3, and 7.5 pounds. Find the fourth weight, x: $6 \le \frac{5.2+6.3+7.5+x}{4} \le 8$. This is $24 \le 19+x \le 32$, so the baby can weigh $5 \le x \le 13$ pounds.

How can we use inequalities to find a missing data point for a desired average?

Example Problem

For a student to earn a B, their average on 4 tests must be between 80 and 89, inclusive. Their first three test scores are 78, 85, and 81. What score is needed on the fourth test?

Step-by-Step

1. First, set up a compound inequality to represent the situation. Let $x$ be the score on the fourth test.

2. Simplify the sum of the known scores in the numerator.

3. To clear the fraction, multiply all three parts of the inequality by $4$.

4. Simplify the multiplication in each part.

5. To isolate $x$, subtract $244$ from all three parts of the inequality.

6. Perform the subtraction to find the range of possible scores for the fourth test.

The student must score between a 76 and a 112 on the fourth test.