Question

2 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 orexplain your reasoning.

Explanation:

Step1: Analyze the equation and inequality context

Since the problem is about comparing costs of two options (implied by the teacher's actions of writing cost - related equations and inequalities), \(n\) likely represents the number of items (a common variable in cost - quantity relationships).

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

The left - hand side \(9(n + 3)\) can be thought of as the cost formula for one option (maybe a base cost of \(9\times3\) and \(9\) per item \(n\)). The right - hand side \(10(n + 1)\) is the cost formula for the second option (a base cost of \(10\times1\) and \(10\) per item \(n\)). The equation means that the cost of the first option (with the formula \(9(n + 3)\)) is equal to the cost of the second option (with the formula \(10(n + 1)\)).

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

Solve the inequality:

$$
LATEXBLOCK0
$$

This means that when the number of items \(n\) is greater than \(17\), the cost of the first option (with the formula \(9(n + 3)\)) is less than the cost of the second option (with the formula \(10(n + 1)\)).

Step4: Graph the solution \(n > 17\) on the number line

Draw a number line. Put an open circle at \(17\) (because \(n = 17\) is not included in the solution, since the inequality is strict \(n>17\)) and draw an arrow to the right of \(17\) to represent all real numbers greater than \(17\).

Answer:

a. \(n\) represents the number of items.
b. The equation \(9(n + 3)=10(n + 1)\) means that the cost of the first option (with cost formula \(9(n + 3)\)) is equal to the cost of the second option (with cost formula \(10(n + 1)\)).
c. The solution \(n>17\) of the inequality \(9(n + 3)<10(n + 1)\) means that when the number of items \(n\) is greater than \(17\), the cost of the first option (with formula \(9(n + 3)\)) is less than the cost of the second option (with formula \(10(n + 1)\)).
d. On the number line, draw an open circle at \(17\) and an arrow to the right of \(17\).