Dezeree received a $100 gift card and needs new sweaters for school. The sweaters cost $27.50 each. Write an equation to determine how many sweaters she can buy and spend exactly $100, solve it, and determine if the value is a viable solution.(4 points)

To determine how many sweaters Dezeree can buy with her $100 gift card, we can let \( x \) represent the number of sweaters. Since each sweater costs $27.50, we can write the equation to represent the total cost:

\[
27.50x = 100
\]

Next, we will solve for \( x \):

1. Divide both sides by 27.50:
\[
x = \frac{100}{27.50}
\]

2. Calculate the division:
\[
x = \frac{100}{27.50} = 3.6363...
\]

Since \( x \) must be a whole number (as she can't buy a fraction of a sweater), we will round \( x \) down to the nearest whole number, which is 3.

To check if buying 3 sweaters is viable, we will calculate the total cost of 3 sweaters:

\[
\text{Total cost for 3 sweaters} = 27.50 \times 3 = 82.50
\]

Since the total cost of 3 sweaters is $82.50, which is under $100, this is a viable solution.

We can also check how many sweaters Dezeree can buy for the maximum integer value less than or equal to \( 3.6363... \):

If she were to buy 4 sweaters, the cost would be:

\[
\text{Total cost for 4 sweaters} = 27.50 \times 4 = 110.00
\]

As $110.00 exceeds her gift card limit of $100, she cannot buy 4 sweaters.

**Conclusion:**
Dezeree can buy a maximum of **3 sweaters**, spending **$82.50**, which is a viable solution.