Question

to help her compare the cost of her two options, the teacher first writes the equation 9(n + 3) = 10(n + 1), and then she writes the inequality 9(n + 3) < 10(n + 1).
a. what does n represent in each statement?
b. in this situation, what does the equation 9(n + 3) = 10(n + 1) mean?
c. what does the solution to the inequality 9(n + 3) < 10(n + 1) tell us?
d. graph the solution to the inequality on the number line. be prepared to show or explain your reasoning.

Explanation:

Step1: Determine the meaning of \(n\)

Since the teacher is comparing the cost of two options, \(n\) represents the number of items (or a quantity related to the cost - comparison context).

Step2: Analyze the equation \(9(n + 3)=10(n + 1)\)

The left - hand side \(9(n + 3)\) could represent the cost of one option (where the base cost per unit is \(9\) and there is an additional cost factor of \(3\) per \(n\) items). The right - hand side \(10(n + 1)\) could represent the cost of another option (where the base cost per unit is \(10\) and there is an additional cost factor of \(1\) per \(n\) items). The equation \(9(n + 3)=10(n + 1)\) means that the cost of the two options is equal.

Step3: Analyze the inequality \(9(n + 3)<10(n + 1)\)

The solution to the inequality \(9(n + 3)<10(n + 1)\) tells us the values of \(n\) for which the cost of the first option (represented by \(9(n + 3)\)) is less than the cost of the second option (represented by \(10(n + 1)\)).

Step4: Solve the inequality \(9(n + 3)<10(n + 1)\)

Expand both sides:

$$
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$$

To graph \(n > 17\) on the number line:

• Draw a number line.
• Put an open circle at \(n = 17\) (because \(n = 17\) is not included in the solution, since the inequality is strict \(n>17\)).
• Shade the part of the number line to the right of \(n = 17\).

Answer:

a. \(n\) represents the number of items (or a relevant quantity for cost - comparison).
b. The equation means that the cost of the two options is equal.
c. The solution tells us the values of \(n\) for which the first option's cost is less than the second option's cost.
d. Graph: Open circle at \(n = 17\) and shade to the right.