Question

objective: a - ced.1 create equations and inequalities in one variable and use them to solve problems, include equations arising from linear and quadratic functions, and simple rational and exponential functions.*

symbol	description
>	greater than<br>\is more than\<br>\exceeds\

| ≤ | less than or equal to<br>\at most\<br>
ot greater than\<br>\does not exceed (get over)\<br>\is not more than\ |

≥	greater than or equal to<br>\at least\<br>\is not less than\<br>\is not under\<br>\has minimum value\

david adopts a dog from a shelter. he pays a one - time adoption fee, a veterinary deposit, and a flat fee of $100 for supplies.

• the veterinary deposit is 1½ times the adoption fee.
• david pays a total of $450 to adopt the dog.

how much is the adoption fee?
1.

sarah buys a new television with a down payment, a processing fee, and a flat fee of $50 for delivery.

• the processing fee is ¼ of the down payment.
• sarah pays a total of $950 upfront.

how much is the down payment?
2.

Explanation:

Problem 1 (Adoption Fee for David)

Step 1: Define Variables

Let \( x \) be the adoption fee (in dollars). The veterinary deposit is \( 1\frac{1}{2}x=\frac{3}{2}x \) dollars, and the supply fee is $100. The total cost is $450.

Step 2: Set Up Equation

The sum of adoption fee, veterinary deposit, and supply fee equals total cost:
\( x+\frac{3}{2}x + 100=450 \)

Step 3: Combine Like Terms

Combine \( x \) and \( \frac{3}{2}x \): \( \frac{2}{2}x+\frac{3}{2}x=\frac{5}{2}x \). So the equation becomes \( \frac{5}{2}x + 100 = 450 \).

Step 4: Solve for \( x \)

Subtract 100 from both sides: \( \frac{5}{2}x=450 - 100 = 350 \).
Multiply both sides by \( \frac{2}{5} \): \( x = 350\times\frac{2}{5}=140 \).

Step 1: Define Variables

Let \( y \) be the down payment (in dollars). The processing fee is \( \frac{1}{4}y \) dollars, and the delivery fee is $50. The total upfront cost is $950.

Step 2: Set Up Equation

The sum of down payment, processing fee, and delivery fee equals total upfront cost:
\( y+\frac{1}{4}y + 50=950 \)

Step 3: Combine Like Terms

Combine \( y \) and \( \frac{1}{4}y \): \( \frac{4}{4}y+\frac{1}{4}y=\frac{5}{4}y \). So the equation becomes \( \frac{5}{4}y + 50 = 950 \).

Step 4: Solve for \( y \)

Subtract 50 from both sides: \( \frac{5}{4}y=950 - 50 = 900 \).
Multiply both sides by \( \frac{4}{5} \): \( y = 900\times\frac{4}{5}=720 \).

Answer:

The adoption fee is \(\$140\).

Problem 2 (Down Payment for Sarah)