To determine how many T-shirts Carlotta must sell to make a profit, we need to compare her total production costs to her total money earned from selling the shirts. Let's break down the given equations and find when the money earned exceeds the production costs:
- Production Cost Equation: [tex]\( c = 28 + 3s \)[/tex]
- Money Earned Equation: [tex]\( m = 10s \)[/tex]
Where [tex]\( c \)[/tex] is the total cost, [tex]\( m \)[/tex] is the total money earned, and [tex]\( s \)[/tex] is the number of shirts.
To make a profit, the money earned ([tex]\( m \)[/tex]) must be greater than the production cost ([tex]\( c \)[/tex]). Therefore, we need:
[tex]\[ m > c \][/tex]
Substituting the given equations into this inequality, we have:
[tex]\[ 10s > 28 + 3s \][/tex]
Next, isolate [tex]\( s \)[/tex] by performing algebraic operations:
1. Subtract [tex]\( 3s \)[/tex] from both sides of the inequality:
[tex]\[ 10s - 3s > 28 \][/tex]
2. Simplify the left side:
[tex]\[ 7s > 28 \][/tex]
3. Divide both sides by 7 to solve for [tex]\( s \)[/tex]:
[tex]\[ s > 28 / 7 \][/tex]
[tex]\[ s > 4 \][/tex]
This inequality tells us that Carlotta must sell more than 4 shirts to make a profit. Therefore, the smallest integer number of shirts she must sell to achieve a profit is 5.
Thus, Carlotta must sell at least 5 shirts to make a profit.
