Example Card: Solving a Conjunction (Phone Bill)
Solve conjunction inequalities representing AND compound conditions in Grade 9 Algebra. Write solution sets in interval notation and graph on a number line.
Key Concepts
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. $$45 15 \le 15 + 0.20x 15 \le 65 15$$ $$30 \le 0.20x \le 50$$ 3. Now, divide each part by $0.20$ to isolate $x$. $$\frac{30}{0.20} \le \frac{0.20x}{0.20} \le \frac{50}{0.20}$$ $$150 \le x \le 250$$.
Common Questions
What is a conjunction inequality?
A conjunction inequality is a compound inequality using AND, requiring both conditions to be true simultaneously. The solution set is the intersection of both solution sets, typically a bounded interval between two values on the number line.
How do you solve and graph a conjunction inequality?
Solve each part of the AND inequality separately, then find the overlap of the two solution sets. Graph both inequalities on the same number line and shade the region where they intersect, using closed or open circles as appropriate.
What is a real-world example of a conjunction inequality?
A phone bill problem might require that you spend at least $20 but no more than $50, written as 20 ≤ x ≤ 50. This is a conjunction because both conditions—minimum and maximum—must hold at the same time.