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 2000 - 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.

Explanation:

Step1: Define the inequality

The number of T - shirts sold is given by the expression \(2000 - 40t\), and we want this number to be at least 1200. "At least" means greater than or equal to. So the inequality is \(2000-40t\geq1200\).

Step2: Subtract 2000 from both sides

To isolate the term with \(t\), we subtract 2000 from both sides of the inequality:
\(2000 - 40t-2000\geq1200 - 2000\)
Simplifying both sides, we get \(- 40t\geq - 800\).

Step3: Divide both sides by - 40 (and reverse the inequality)

When we divide both sides of an inequality by a negative number, the direction of the inequality sign changes. So we divide both sides by - 40:
\(\frac{-40t}{-40}\leq\frac{-800}{-40}\)
Simplifying, we get \(t\leq20\). Also, since the price of a T - shirt can't be negative (in a real - world context), we also have \(t > 0\). But from the inequality we solved, the main inequality gives us \(t\leq20\). Combining with the non - negativity of price, the range of \(t\) is \(0 < t\leq20\). But if we just consider the inequality from the number of T - shirts:

The inequality \(2000-40t\geq1200\) is solved as follows:

Starting with \(2000-40t\geq1200\)

Subtract 2000: \(-40t\geq1200 - 2000=-800\)

Divide by - 40 (reverse inequality): \(t\leq\frac{-800}{-40}=20\)

Answer:

The inequality is \(2000 - 40t\geq1200\), and the solution for \(t\) (the price of a T - shirt in dollars) is \(t\leq20\) (and \(t>0\) for real - world price context). If we just consider the inequality solution from the given relationship, \(t\leq20\).