The telephone company offers two billing plans for local calls.  Plan 1 charges ​$35 per month for unlimited calls and plan 2 charges ​$13 per month plus 5 cents per call.  Use an inequality to find the number of monthly calls for which plan 1 is more economical than plan 2.


Let's set up the inequality to compare the two plans.

### Plan 1:
- Charges $\$35$ per month regardless of the number of calls.

### Plan 2:
- Charges $\$13$ per month plus 5 cents per call.

Let $ n $ represent the number of calls made in a month.

For Plan 2, the cost is given by:
\[
\text{Cost of Plan 2} = 13 + 0.05n
\]

We want to find when Plan 1 is more economical than Plan 2, meaning the cost of Plan 1 should be less than the cost of Plan 2:
\[
35 < 13 + 0.05n
\]

### Solving the Inequality:

1. Subtract 13 from both sides:
\[
35 - 13 < 0.05n
\]
\[
22 < 0.05n
\]

2. Divide both sides by 0.05 to solve for $ n $:
\[
\frac{22}{0.05} < n
\]
\[
440 < n
\]

### Conclusion:
Plan 1 is more economical than Plan 2 when the number of calls $ n $ exceeds 440 calls in a month.

Would you like any details or have any questions?

Here are 5 related questions:
1. How do we solve the inequality if the roles of the plans were reversed?
2. What if Plan 2 charges 7 cents per call instead of 5 cents?
3. How would you model the situation if Plan 2 also had a free call allowance before charges begin?
4. What happens if both plans include a discount after a certain number of calls?
5. How can this inequality be represented graphically?

**Tip:** When comparing costs, always set up an inequality that represents when one option is cheaper than the other, then solve for the variable in question.

Understand how to compare two telephone billing plans using inequalities. Learn when Plan 1 becomes more economical than Plan 2 based on the number of monthly calls. Explore step-by-step solutions and mathematical reasoning. Suitable for Grades 7-9.

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