Sal is trying to determine which cell phone and service plan to buy for his mother. The first phone costs $100 and $55 per month

Answer:

Part 1:

The initial phone has a cost of $100 and a monthly charge of $55 for unlimited usage.

Let f(x) denote the cost of the first phone, with x representing the months.

Equation forms:

The second phone costs $150, with a monthly rate of $51 for unlimited service.

Let g(x) denote the cost of the second phone, with x as the number of months.

Equation forms:

We need to establish the inequality that shows the months, x, required for the second phone's cost to be lesser. This is given by:

Part 2:

The inequality solution is:  

=>

=>

=>

=>

Or rounded to 13 months.

Part 3:

For the second phone plan to be less costly, Sal's mother needs to keep it for no less than 13 months.

Answer:

a) The first inequality is 100 + 55x > 150 + 51x;

b) The final inequality results in x > 12.5

c) Sal's mother will need to use the second phone for at least 13 months.

Step-by-step explanation:

a) Let x represent the number of months.

1. The first phone is priced at $100, with a monthly fee of $55 for unlimited use, leading to a total cost of $(100 + 55x) for x months.

2. The second phone costs $150 with a monthly fee of $51 for unlimited use, resulting in a total of $(150 + 51x) for x months.

3. For the second phone to be cheaper, we set up the inequality:

150 + 51x < 100 + 55x

which simplifies to

100 + 55x > 150 + 51x

b) Now solve this:

55x - 51x > 150 - 100

4x > 50

so x > 12.5

c) This means Sal's mother has to retain the second phone for at least 13 months (since x > 12.5).