Question

(PLEASE HELP ME DX PLEASE) Nate would like to purchase a new laptop for school from Best Buy. The store is offering a $100 mail in rebate on all Lenovo Laptops. Nate is planning to spend anywhere between $300 - $500. Sales tax is 8% of the total cost after the rebate has been given.

(a) Write the compound inequality to represent this scenario.

(b) What price range of laptops can Nate look at so that he meets his budget. Show your work. Be sure to
explain what your answer means.

Answer 1

This would be the compound inequality formula for the given situation. This is because Nate is planning on spending somewhere between 300 and 500 dollars. Therefore the price (represented by the variable p) needs to be greater than or equal to 300 but at the same time less than or equal to 500. But this is after the rebate and sales tax have been calculated into the laptop's price. Which in this scenario, the rebate needs to be applied first and then the sales tax is applied after as 1.08 in order to apply the 8% onto the price itself.

Answer 2

Part A)

`300\leq 1.08(x-100)\leq 500`

Part B)

`377.78\leq x\leq 562.96`

Nate wants the original price of the laptop to be between approximately $378 or $563.

Explanation:

Part A)

Let's let x be the price of the new laptop.

Since the store is offering a $100 rebate (discount), this means that the total cost of the laptop before tax is:

`x-100`

So, the total price of the laptop after the 8% tax is:

`1.08(x-100)`

Since Nate is planning to spend between $300 and $500, this means that we can write the following compound inequality:

`300\leq 1.08(x-100)\leq 500`

Part B)

To find the range of the price of laptops, let's find the solution to our inequality.

To do so, let's solve them individually. So, let's first find our minimum price:

`300\leq1.08(x-100)`

Divide both sides by 1.08:

`277.78\leq x-100`

Add 100 to both sides:

`377.78\leq x`

So, Nate wants the laptop to be at least approximately $378.00.

Now, let's find our maximum price:

`1.08(x-100)\leq 500`

Divide both sides by 1.08:

`x-100\leq462.96`

Add 100 to both sides:

`x\leq 562.96`

So, Nate wants the maximum price of the laptop to be about $563.00.

So, our compound inequality is:

`377.78\leq x\leq 562.96`

This means that Nate wants the original price of the laptop to be between approximately $378 or $563.

And we're done!