Grayson charges $35 per hour plus a $35 administration fee for tax preparation. Ian charges $45 per hour plus a $15 administration fee. If [tex]$h$[/tex] represents the number of hours of tax preparation, for what number of hours does Grayson charge more than Ian?

A. [tex]$h\ \textless \ 2$[/tex]
B. [tex]$h\ \textgreater \ 2$[/tex]
C. [tex]$h\ \textless \ 5$[/tex]
D. [tex]$h\ \textgreater \ 5$[/tex]

Grayson charges more than Ian when the number of hours $h$ is less than 2 hours, which corresponds to option A. Therefore, when $h < 2$, Grayson is the more expensive choice for tax preparation. This conclusion stems from setting up and solving an inequality based on their respective charges.
;

Answered by Anonymous | 2025-07-04

Express Grayson's charge as G ( h ) = 35 h + 35 and Ian's charge as I ( h ) = 45 h + 15 .
Set up the inequality 45h + 15"> 35 h + 35 > 45 h + 15 to find when Grayson charges more than Ian.
Solve the inequality: 10h"> 20 > 10 h , which simplifies to h < 2 .
Conclude that Grayson charges more than Ian when h < 2 . h < 2 ​

Explanation

Understanding the Problem Let's analyze the problem. We are given that Grayson charges $35 per hour plus a $35 administration fee, and Ian charges $45 per hour plus a $15 administration fee. We want to find the number of hours, h , for which Grayson's charge is more than Ian's charge.

Expressing the Charges as Functions Let G ( h ) be the total charge for Grayson and I ( h ) be the total charge for Ian. We can express these charges as functions of the number of hours, h :


Grayson's charge: G ( h ) = 35 h + 35 Ian's charge: I ( h ) = 45 h + 15

Setting up the Inequality We want to find the values of h for which Grayson charges more than Ian, so we set up the inequality:

I(h)"> G ( h ) > I ( h ) 45h + 15"> 35 h + 35 > 45 h + 15

Solving the Inequality Now, let's solve the inequality for h :

45h + 15"> 35 h + 35 > 45 h + 15 Subtract 35 h from both sides: 10h + 15"> 35 > 10 h + 15 Subtract 15 from both sides: 10h"> 20 > 10 h Divide both sides by 10 :
h"> 2 > h So, h < 2

Conclusion Therefore, Grayson charges more than Ian when the number of hours, h , is less than 2.

Examples
Understanding when one service becomes more cost-effective than another is a common real-world problem. For instance, consider choosing between two internet providers: one with a lower monthly fee but higher installation costs, and another with a higher monthly fee but lower installation costs. By setting up equations and inequalities similar to this tax preparation problem, you can determine the number of months you would need to subscribe for one provider to become cheaper than the other. This type of analysis helps in making informed decisions based on usage and cost over time.

Answered by GinnyAnswer | 2025-07-04