Step-by-step explanation:
Let's represent the given situation with the following system of inequalities:
1. Jayden has a maximum of 22 coins: \( x + y \leq 22 \) (since each nickel or dime represents one coin).
2. The combined value of the coins is at least $1.40: \( 0.05x + 0.10y \geq 1.40 \) (nickels are worth $0.05 and dimes are worth $0.10).
Let's graph these inequalities to find a possible solution.
First, we'll graph the line \( x + y = 22 \) and shade the area below or on the line. Then, we'll graph the line \( 0.05x + 0.10y = 1.40 \) and shade the area above or on the line.
The graph will show the feasible region that satisfies both inequalities, and the intersection of these shaded areas will represent the possible solutions.
Let's proceed with plotting these lines and finding a possible solution.
Apologies, as a text-based AI, I can't directly generate graphical plots. However, I can guide you through the process of solving the system of inequalities.
First, rearrange the inequalities to solve for y in terms of x:
1. \( x + y \leq 22 \) can be rewritten as \( y \leq 22 - x \).
2. \( 0.05x + 0.10y \geq 1.40 \) can be rewritten as \( 0.10y \geq 1.40 - 0.05x \) or \( y \geq \frac{1.40 - 0.05x}{0.10} \).
Now you have the equations in slope-intercept form (\(y = mx + b\)):
1. \( y \leq 22 - x \)
2. \( y \geq 14 - 0.5x \)
Graph these two inequalities on the same coordinate plane. The area that satisfies both conditions represents the feasible region.
Look for the intersection of these shaded regions to find a possible solution. This point represents a combination of nickels (\(x\)) and dimes (\(y\)) that meets the given conditions.
