You recently started the paperwork to purchase your new home, and were just notified that you can move into the house in 2 weeks. You decide to hire a moving company, but are unsure which company to choose. You search online and are interested in contacting two companies, Heavy Lifting and Quick Move, to discuss their rates. Heavy Lifting charges a $75 fee plus $50 per hour. Quick Move charges $100 for the 1st hour and $60 for each additional hour.

a. Which mover provides a better deal for 2 hours of work? How did you arrive at your answer?

b. Which mover provides a better deal for 12 hours of work? How did you arrive at your answer?

c. For what values h (hours) does Quick Move offer the better deal? Express your answer as an inequality. Explain how you reached your answer.

a. To determine which mover provides a better deal for 2 hours of work, we can compare the total cost for both companies.

For Heavy Lifting:
- There is a $75 fee plus $50 per hour, so for 2 hours, the cost would be $75 + ($50 * 2) = $175.

For Quick Move:
- They charge $100 for the 1st hour and $60 for each additional hour, so for 2 hours, the cost would be $100 + $60 = $160.

Comparing the two, Quick Move offers a better deal for 2 hours of work since their cost is $160, which is lower than Heavy Lifting's cost of $175.

b. To determine which mover provides a better deal for 12 hours of work, we can compare the total cost for both companies.

For Heavy Lifting:
- They charge a $75 fee plus $50 per hour, so for 12 hours, the cost would be $75 + ($50 * 12) = $675.

For Quick Move:
- They charge $100 for the 1st hour and $60 for each additional hour, so for 12 hours, the cost would be $100 + ($60 * 11) = $760.

Comparing the two, Heavy Lifting offers a better deal for 12 hours of work since their cost is $675, which is lower than Quick Moves's cost of $760.

c. To determine for what values of hours h Quick Move offers the better deal, we can set up an inequality.

Let's assume x represents the number of additional hours (excluding the first hour). So, h = x + 1.

For Quick Move to offer the better deal, the cost of Quick Move should be lower than Heavy Lifting for any value of h.

The cost for Quick Move is $100 (for the first hour) + $60x (for additional hours)
The cost for Heavy Lifting is $75 (fee) + $50h (for hours)

Therefore, we need to find the values of h for which the cost of Quick Move is lower than the cost of Heavy Lifting:

$100 + $60x < $75 + $50h

Substituting h = x + 1:
$100 + $60x < $75 + $50(x + 1)

Simplifying the expression:
$100 + $60x < $75 + $50x + $50
$60x - $50x < $75 + $50 - $100
$10x < $25

Dividing both sides of the inequality by 10:
x < 2.5

Therefore, for values of h less than 2.5, Quick Move offers the better deal. As a whole number, this means Quick Move offers the better deal for 2 hours of work.