Question

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 for unlimited usage. The second phone costs $150 and $51 per month for unlimited usage. How many months will it take for the second phone to be less expensive than the first phone?

The inequality that will determine the number of months, x, that are required for the second phone to be less expensive is 100 + 55x > 150 + 51x100 + 55x < 150 + 51x100x+ 55 > 150x+ 51100x+ 55 < 150x+ 51.

The solution to the inequality is x > 2.4x < 2.4x < 12.5x > 12.5.

Sal’s mother would have to keep the second cell phone plan for at least 231213 months in order for it to be less expensive​

Answer 1

Part 1:

The first phone costs $100 and $55 per month for unlimited usage.

Let f(x) be the cost of the first phone and x be the number of months.

Equation forms:

`f(x)=55x+100`

The second phone costs $150 and $51 per month for unlimited usage.

Let g(x) be the cost of the second phone and x be the number of months.

Equation forms:

`g(x)=51x+150`

We have to find the inequality that will determine the number of months, x, that are required for the second phone to be less expensive, it is given by:

`g(x)<f(x)`

`51x+150<55x+100`

Part 2:

The solution to the inequality is:

`51x+150<55x+100`

=>
`51x-55x<100-150`

=>
`-4x<-50`

=>
`-x<-12.5`

=>
`x>12.5`

Or rounding off to 13.

Part 3:

Sal’s mother would have to keep the second cell phone plan for at least 13 months in order for it to be less expensive.

Answer 2

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

b) The last inequality x>12.5

c) 13 months

Explanation:

a) Let x be the number of months.

1. The first phone costs $100 and $55 per month for unlimited usage, then for x months it will cost $55x and in total

$(100+55x)

2. The second phone costs $150 and $51 per month for unlimited usage, then for x months it will cost %51x and in total

$(150+51x)

3. If the second phone must be less expensive than the first phone, then

150+51x<100+55x

or

100+55x>150+51x

b) Solve this inequality:

55x-51x>150-100

4x>50

x>12.5

c) Sal's mother has to keep the second cell phone for at least 13 months (because x>12.5).