$15$ . You are buying plants and soil for your garden. The soil costs $\$ 4.00$ per bag and the plants cost $\$ 10.00$ each. You want to buy at least $5$ plants and can spend no more than $\$ 100$ total. $\newline$ a. Write a system of linear inequalities to model the situation. $\newline$ b. Graph the system of linear inequalities.

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15

. You are buying plants and soil for your garden. The soil costs

\$ 4.00

per bag and the plants cost

\$ 10.00

each. You want to buy at least

5

plants and can spend no more than

\$ 100

total.

\newline

a. Write a system of linear inequalities to model the situation.

\newline

b. Graph the system of linear inequalities.

Representing Variables: Let $x$ represent the number of bags of soil and $y$ represent the number of plants.
Calculating Costs: The soil costs $\$4$ per bag, so $4x$ represents the total cost of the soil.
Setting Inequalities: The plants cost $\$10$ each, so $10y$ represents the total cost of the plants.
Graphing Inequalities: You want to buy at least $5$ plants, which gives us the inequality $y \geq 5$ .
Finding Intercepts: You can spend no more than $\$100$ total, so the combined cost of soil and plants should be less than or equal to $\$100$ : $4x + 10y \leq 100$ .
Plotting Points: The system of linear inequalities is: $\newline$ $1$ . $y \geq 5$ $\newline$ $2$ . $4x + 10y \leq 100$
Shading Regions: To graph the inequalities, first graph the line $4x + 10y = 100$ .
Determining Solution: Find the $x$ -intercept by setting $y$ to $0$ : $4x = 100$ , so $x = 25$ .
Determining Solution: Find the $x$ -intercept by setting $y$ to $0$ : $4x = 100$ , so $x = 25$ . Find the $y$ -intercept by setting $x$ to $0$ : $10y = 100$ , so $y = 10$ .
Determining Solution: Find the x-intercept by setting $y$ to $0$ : $4x = 100$ , so $x = 25$ .Find the y-intercept by setting $x$ to $0$ : $10y = 100$ , so $y = 10$ .Plot the points $(25, 0)$ and $(0, 10)$ and draw the line.
Determining Solution: Find the x-intercept by setting $y$ to $0$ : $4x = 100$ , so $x = 25$ . Find the y-intercept by setting $x$ to $0$ : $10y = 100$ , so $y = 10$ . Plot the points $(25, 0)$ and $(0, 10)$ and draw the line. Since the inequality is $0$ $0$ , shade the area below the line.
Determining Solution: Find the $x$ -intercept by setting $y$ to $0$ : $4x = 100$ , so $x = 25$ .Find the $y$ -intercept by setting $x$ to $0$ : $10y = 100$ , so $y = 10$ .Plot the points $y$ $0$ and $y$ $1$ and draw the line.Since the inequality is $y$ $2$ , shade the area below the line.Graph the inequality $y$ $3$ by drawing a horizontal line through $y$ $4$ and shading above the line.
Determining Solution: Find the x-intercept by setting $y$ to $0$ : $4x = 100$ , so $x = 25$ .Find the y-intercept by setting $x$ to $0$ : $10y = 100$ , so $y = 10$ .Plot the points $(25, 0)$ and $(0, 10)$ and draw the line.Since the inequality is $0$ $0$ , shade the area below the line.Graph the inequality $0$ $1$ by drawing a horizontal line through $0$ $2$ and shading above the line.The solution to the system of inequalities is the shaded region that overlaps both conditions on the graph.