Lesson 73: Solving Compound Inequalities

New Concept

A compound inequality is two inequalities combined with the word AND or OR.

What’s next

Next, you’ll apply this concept to write, graph, and solve problems involving both 'AND' (conjunctions) and 'OR' (disjunctions) statements.

Property

A compound inequality consists of two inequalities joined by the word AND or OR. A conjunction is a compound inequality linked by AND. For instance, the statement $x \ge -3$ AND $x \le 5$ can be written more compactly as $-3 \le x \le 5$.

Explanation

Think of a conjunction as a VIP club with two rules: you must be old enough AND have an invitation. You have to satisfy both conditions to get in! The solution is the 'sweet spot' where both inequalities overlap, representing all the numbers that fit both criteria at the same time.

Examples

• All real numbers greater than 2 and less than 7 are written as $2 < x < 7$.
• A recommended lasagna cooking time is from 16 to 20 minutes: $16 \le t \le 20$.
• A graph showing a line segment between solid dots at -2 and 6 is represented by $-2 \le x \le 6$.

Property

A conjunction can be solved as two inequalities connected with AND or as a single inequality with three parts, such as $30 \le 10 + 0.10x \le 40$.

Explanation

Solving a three-part inequality is like a balancing act! To find your variable, you must perform the same operation on all three parts—the left, the middle, and the right. This ensures that you keep the relationship between the numbers fair and balanced while you work to isolate the variable in the center.

Examples

• A phone bill between 30 and 40 dollars, with a 10 dollars fee and 0.10 dollars per minute, $x$, is $30 \le 10 + 0.10x \le 40$.
• First, subtract 10 from all three parts: $20 \le 0.10x \le 30$.
• Next, divide all three parts by 0.10 to find the range of minutes: $200 \le x \le 300$.

Let's see how compound inequalities can help you stay within your monthly phone budget. This example uses a conjunction, where two conditions must be true.

Example Problem: A phone plan costs 15 dollars plus $0.20 per minute. The monthly bill is between 45 dollars and 65 dollars. Find the possible minutes used.

Step-by-Step:

1. Let $x$ be the number of minutes. The total cost is $15 + 0.20x$. The bill is between 45 dollars and 65 dollars, so we write the conjunction: $45 \le 15 + 0.20x \le 65$.
2. We solve this as one inequality. First, subtract 15 from all three parts.

3. Now, divide each part by $0.20$ to isolate $x$.

Property

A disjunction is a compound inequality that uses the word OR. A solution to a disjunction is any number that makes either of the inequalities true. These must be written as two separate inequalities connected by OR.

Explanation

Think of a disjunction as a choice at a theme park: you can ride the roller coaster OR the Ferris wheel. You don't have to do both! Any number that satisfies at least one of the conditions is part of the solution, which is why the graphs often shoot off in opposite directions.

Examples

• All real numbers greater than 9 or less than 3 is written as $x < 3$ OR $x > 9$.
• All real numbers no more than 7 or no less than 11 is written as $x \le 7$ OR $x \ge 11$.
• The solution to $x - 3 < -5$ OR $-2x < -6$ is $x < -2$ OR $x > 3$.

What if only one of two conditions needs to be true? This example uses a disjunction, connected by 'OR'.

Example Problem: Solve the disjunction $x + 2 < -4$ OR $-3x < -12$.

Step-by-Step:

1. Solve each inequality separately. For the first one, subtract 2 from both sides.

2. For the second inequality, divide by $-3$. Remember to reverse the inequality sign when dividing by a negative number.

3. Combine the two results with "OR". The solution is any number that satisfies either condition: $x < -6$ OR $x > 4$.