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You are looking to get cable internet service in your new apartment. Charter charges [tex]$60[/tex] per month for its internet service along with a [tex]\$50[/tex] installation charge. They are running a special where you can bundle internet and TV together for [tex]\$80[/tex] per month along with free installation.

Write an inequality to show how long it will take for the bundled price to be more expensive than the internet-only price.

A. [tex]80x \leq 60x + 50[/tex]
B. [tex]80x \geq 60x + 50[/tex]
C. [tex]80 \leq 60 + 50x[/tex]
D. [tex]80 \geq 60 + 50x[/tex]

Answer :

Certainly! Let's break down the situation and solve it step by step to find out how long it will take for the bundled price to become more expensive than just the internet-only service.

Step-by-step Solution:

1. Identify the Costs:
- Internet-Only Plan:
- Monthly Cost: [tex]$60
- One-Time Installation Charge: $[/tex]50

- Bundle Plan (Internet + TV):
- Monthly Cost: [tex]$80
- Installation Charge: $[/tex]0 (Free)

2. Set Up the Inequality:
We want to find when the total cost of the bundled plan becomes more expensive than the internet-only plan.

For the internet-only plan, the total cost after [tex]\(x\)[/tex] months is:
[tex]\[
60x + 50
\][/tex]

For the bundle plan, the total cost after [tex]\(x\)[/tex] months is:
[tex]\[
80x
\][/tex]

We need to find when the bundle's cost is greater than the internet-only plan's cost:
[tex]\[
80x > 60x + 50
\][/tex]

3. Solve the Inequality:
Subtract [tex]\(60x\)[/tex] from both sides to isolate terms with [tex]\(x\)[/tex]:
[tex]\[
80x - 60x > 50
\][/tex]

Simplify the inequality:
[tex]\[
20x > 50
\][/tex]

Divide both sides by 20 to solve for [tex]\(x\)[/tex]:
[tex]\[
x > \frac{50}{20}
\][/tex]

Simplify:
[tex]\[
x > 2.5
\][/tex]

4. Interpret the Result:
Since [tex]\(x\)[/tex] represents the number of months, and we are looking for whole months, the bundling plan becomes more expensive after more than 2.5 months. This means that after 3 months, the cost of the bundled service will exceed the cost of just having internet service alone.

Thus, starting from month 3, the bundled plan is more costly than sticking with just the internet service.

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