Question

a store owner wants to sell at least 1,200 t - shirts. if the price of one t - shirt is t dollars, the number of t - shirts the store owner will sell is 2,000 - 40t. what range of prices will result in the store owner selling at least 1,200 t - shirts?
a. write and solve an inequality to model the situation.
b. graph the solution set of the inequality on the number line.
(there is a number line with arrows on both ends and several vertical marks on it)
c. determine the range of prices that will result in sales of at least 1,200 t - shirts. consider restrictions to the solution set based on the context. explain your thinking.

Explanation:

Part a

Step1: Define the inequality

The number of T - shirts sold is \(2000 - 40t\) and we want to sell at least 1200 T - shirts. So the inequality is \(2000-40t\geq1200\).

Step2: Subtract 2000 from both sides

Subtract 2000 from each side of the inequality: \(2000 - 40t-2000\geq1200 - 2000\), which simplifies to \(- 40t\geq - 800\).

Step3: Divide by - 40 (reverse inequality)

Divide both sides by - 40. When dividing an inequality by a negative number, the direction of the inequality sign changes. So \(\frac{-40t}{-40}\leq\frac{-800}{-40}\), which gives \(t\leq20\). Also, since the price \(t\) must be non - negative (you can't have a negative price), we also have \(t\geq0\). But we are mainly solving \(2000 - 40t\geq1200\) for now.

To graph the solution set of \(t\leq20\) (and \(t\geq0\)) on the number line:

1. Draw a number line.
2. Locate the point 20 on the number line. Since the inequality is \(t\leq20\), we use a closed circle at 20 (because the inequality includes equality).
3. Then draw an arrow to the left from 20 to represent all numbers less than or equal to 20. Also, since the price can't be negative, we start from 0 (with a closed circle at 0 as well, because \(t = 0\) is a valid price in theory, although in a real - world context, the price is likely to be greater than 0, but from the inequality \(2000-40t\geq1200\) we first get \(t\leq20\) and \(t\geq0\) from the context of price being non - negative).

1. From part a, we have the inequality \(2000-40t\geq1200\) which solves to \(t\leq20\).
2. In the context of the problem, the price of a T - shirt \(t\) must be greater than or equal to 0 (because a negative price doesn't make sense in a real - world selling situation). Also, we can think about the upper and lower bounds. If \(t = 0\), the number of T - shirts sold is \(2000-40(0)=2000\) which is more than 1200. As \(t\) increases, the number of T - shirts sold decreases. When \(t = 20\), the number of T - shirts sold is \(2000-40\times20=2000 - 800 = 1200\), which meets the "at least 1200" requirement. If \(t>20\), say \(t = 21\), then the number of T - shirts sold is \(2000-40\times21=2000 - 840 = 1160<1200\), which does not meet the requirement. And since the price can't be negative, \(t\geq0\). But in a real - world store - owner situation, the price is likely to be greater than 0 (because selling T - shirts for $0$ would not be a sustainable business model), but from the mathematical model and the basic context of non - negative price, the range of prices is \(0\leq t\leq20\) (in dollars).

Answer:

The inequality is \(2000 - 40t\geq1200\), and the solution (before considering non - negative price) is \(t\leq20\).

Part b