A teacher wants to buy 9 boxes of granola bars for a school trip. Each box
usually costs $7, but many grocery stores are having a sale on granola bars
this week. Different stores are selling boxes of granola bars at different
discounts.
a. If x represents the dollar amount of the discount, then the amount the
teacher will pay can be expressed as 9(7-x). In this expression, what
does the quantity 7 - x represent?
b. The teacher has $36 to spend on the granola bars. The equation
9(7-x) = 36 represents a situation where she spends all $36.
Solve this equation.
What does the solution mean in this situation?
d. The teacher does not have to spend all $36. Write an inequality relating
36 and 9(7- x) representing this situation.
The solution to this inequality must either look like x ≥ 3 or x ≤ 3. Whic
do you think it is? Explain your reasoning.

a) In the expression 7 - x, the quantity 7 - x represents the amount of discount applied to each box of granola bars. Since the original price of each box is $7, subtracting x from 7 gives us the reduced price after the discount.

b) To solve the equation 9(7 - x) = 36, we can start by simplifying the left side of the equation:

9(7 - x) = 36

63 - 9x = 36

Subtracting 63 from both sides:

-9x = 36 - 63

-9x = -27

Dividing both sides by -9 (to isolate x):

x = -27 / -9

x = 3

The solution x = 3 means that the discount offered on each box of granola bars is $3. Therefore, the teacher will pay $7 - $3 = $4 for each box. Since the teacher wants to buy 9 boxes, the total cost will be 9 * $4 = $36, which matches the amount she has to spend.

d) The teacher does not have to spend all $36. To represent this situation with an inequality, we can use the symbol ≤ (less than or equal to). The inequality would be:

9(7 - x) ≤ 36

This inequality states that the cost of 9 boxes of granola bars (9 times the discounted price) should be less than or equal to $36. The solution to this inequality will determine the range of possible values for x.

To determine whether the solution is x ≥ 3 or x ≤ 3, we can solve the inequality:

9(7 - x) ≤ 36

63 - 9x ≤ 36

-9x ≤ 36 - 63

-9x ≤ -27

Dividing both sides by -9 (and reversing the inequality):

x ≥ -27 / -9

x ≥ 3

The solution to the inequality is x ≥ 3. This means that the teacher can choose a discount equal to or greater than $3 per box of granola bars, and the total cost will be less than or equal to $36.