Problem 53 - College Algebra

Question

You are choosing between two different prepaid cell phone plans. The first plan charges a rate of 26 cents per minute. The second plan charges a monthly fee of \(\$ 19.95\) plus 11 cents per minute. How many minutes would you have to use in a month in order for the second plan to be preferable?

Step-by-Step Solution

Answer

Use more than 133 minutes for the second plan to be cheaper.

Step 1: Define variables and expressions

Let's define the variable \( x \) to represent the number of minutes used in a month. We'll need to calculate the total cost for each plan based on \( x \). For the first plan, the cost, \( C_1 \), is \( 0.26x \) because it charges 26 cents per minute. For the second plan, the cost, \( C_2 \), is \( 19.95 + 0.11x \), where \\(19.95 is the flat monthly fee and \\)0.11 is the per-minute charge.

Step 2: Set up the inequality

To find out when the second plan becomes cheaper, we set up an inequality where the cost of the second plan is less than the cost of the first plan. The inequality looks like: \( 19.95 + 0.11x < 0.26x \). This means we want to find \( x \) such that the expression for the second plan's cost is less than the first plan's cost.

Step 3: Solve the inequality

Subtract \( 0.11x \) from both sides to isolate terms with \( x \) on one side: \( 19.95 < 0.26x - 0.11x \). Simplify the right side: \( 19.95 < 0.15x \). Now, divide both sides by 0.15 to solve for \( x \): \( x > \frac{19.95}{0.15} \). Calculating the division gives \( x > 133 \).

Step 4: Interpret the solution

This result tells us that you need to use more than 133 minutes in a month for the second plan to be cheaper than the first plan. If you use exactly 133 minutes, the two plans cost the same. Therefore, the second plan is preferable only if your usage exceeds 133 minutes.

Key Concepts

Algebraic Expressions

Algebraic expressions are extremely useful in understanding and solving problems in mathematics, including real-world scenarios like comparing costs.An algebraic expression combines numbers, variables, and operations (like addition or multiplication) to represent a value.In our cell phone plan problem, we use algebraic expressions to calculate the total monthly cost of each plan:

• The first plan's cost is represented as \( C_1 = 0.26x \), which means 26 cents multiplied by the number of minutes used, \( x \).
• The second plan's cost is expressed as \( C_2 = 19.95 + 0.11x \), where \( 19.95 \) is a fixed monthly fee and \( 0.11x \) accounts for the 11 cents per minute charge.

Understanding algebraic expressions helps us form equations and inequalities needed to solve the problem. It involves translating real-world words into mathematical symbols, which makes comparisons more straightforward and accurate.

Cell Phone Plans Comparison

When comparing cell phone plans, it's crucial to consider both fixed and variable costs. A fixed cost is a fee that doesn't change with usage, like the \( 19.95 \) monthly fee in Plan 2. A variable cost fluctuates based on usage, such as the per-minute charges in both plans.Key points to compare cell phone plans effectively:

• Identify all fixed costs (subscription fees, etc.).
• Calculate variable costs based on expected usage (price per minute or per GB).
• Use algebraic expressions to sum fixed and variable costs, allowing easy comparison.

In our exercise, we determine at what point Plan 2 becomes more cost-effective than Plan 1 by evaluating these factors through inequality.

Linear Inequalities

Linear inequalities are a type of algebraic expression that show the relationship between two expressions where one is not equal but rather less than, greater than, less than or equal to, or greater than or equal to the other.In the context of cell phone plans, we use the inequality: \( 19.95 + 0.11x < 0.26x \) to find the tipping point where the second plan is more cost-effective.Key steps in solving a linear inequality:

• Isolate the variable on one side using operations like addition, subtraction, multiplication, or division.
• Rearrange terms to eliminate constants from one side.
• Simplify if needed and solve by division or multiplication to determine the boundary value for the variable.

In this problem, solving the inequality shows that you need to use more than 133 minutes for the second plan to be cheaper than the first. Being able to setup and solve linear inequalities is vital in making informed decisions based on different conditions.