# Janice is going on vacation and needs to leave her dog at a kennel. Nguyen'ts Kennel charges $15/day plus $20 processing fee. The Pup Palace charges $12/day and a $35 processing fee. After how many days is the Pup Palace Kennel cheaper than Nguyen's?

More than 5 days makes Pup Palace the better deal.

To solve this, you have to set up an inequality such that the "Pup Palace" expression is less than the "Nguyen's" expression. To do this, you create small expressions for the cost of each kennel.

Call the number of days "n". There is a fixed portion of the cost for each kennel that does not change based on the number of days, and a cost portion that does.

The price for Pup Palace can be expressed as 12n + 35.

The price for Nguyen's is expressed as 15n + 20.

Create the inequality now such that the Nguyen's expression is greater than the Pup Palace expression:

12n + 35 < 15n + 20 (That is the hardest part to complete!)

We "collect like terms" to solve this. First, subtract 12n from both sides:

35 < 3n + 20

Subtract 20 from both sides now.

15 < 3n

Each side is divided by three.

5 < n

This should be interpreted as n > 5 (is that correct?), which indicates that Pup Palace is the better option for stays longer than five days.

(By the way, you should check that your answer is correct. Six days means Pup Palace costs $107, while Nguyen's is$110.)

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The Pup Palace Kennel becomes cheaper than Nguyen's Kennel after 5 days.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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