Question

1. $-2x + y geq -4$ 17. $8x + y leq 6$ examples 3 - 4 19 mp modeling the girls’ soccer team wants to raise at least $2000 to buy new goals. if they make $1.00 from each hot dog and $1.25 from each soda, how many of each item do they need to sell? a. write an inequality to represent the situation. b. graph an inequality to represent the situation. c. plot at least 5 possible solutions on your graph.

Explanation:

Part (a)

Step1: Define Variables

Let \( x \) be the number of hot dogs sold and \( y \) be the number of sodas sold.

Step2: Determine Earnings

Earnings from hot dogs: \( 1.00x \) dollars. Earnings from sodas: \( 1.25y \) dollars.

Step3: Form Inequality

Total earnings must be at least \( 2000 \) dollars, so \( 1.00x + 1.25y \geq 2000 \).

Step1: Rewrite in Slope - Intercept Form

Start with \( x + 1.25y \geq 2000 \). Solve for \( y \):
\( 1.25y\geq - x + 2000 \)
\( y\geq-\frac{1}{1.25}x+\frac{2000}{1.25} \)
Simplify: \( y\geq - 0.8x + 1600 \)

Step2: Graph the Boundary Line

The boundary line is \( y=-0.8x + 1600 \). Since the inequality is \( \geq \), the line is solid.

Step3: Shade the Region

Test a point not on the line, e.g., \( (0,0) \): \( 0+0\geq2000 \) is false. So shade the region above the line (where \( y\geq - 0.8x + 1600 \)).

We need to find points \( (x,y) \) such that \( x + 1.25y\geq2000 \), \( x\geq0 \), \( y\geq0 \) (since we can't sell a negative number of items).

Step1: Find Points

• When \( x = 0 \): \( 0+1.25y\geq2000\Rightarrow y\geq1600 \). So \( (0,1600) \) is a solution.
• When \( y = 0 \): \( x + 0\geq2000\Rightarrow x\geq2000 \). So \( (2000,0) \) is a solution.
• When \( x = 500 \): \( 500+1.25y\geq2000\Rightarrow1.25y\geq1500\Rightarrow y\geq1200 \). So \( (500,1200) \) (check: \( 500 + 1.25\times1200=500 + 1500 = 2000\geq2000 \))
• When \( x = 1000 \): \( 1000+1.25y\geq2000\Rightarrow1.25y\geq1000\Rightarrow y\geq800 \). So \( (1000,800) \) (check: \( 1000+1.25\times800 = 1000 + 1000=2000\geq2000 \))
• When \( x = 1500 \): \( 1500+1.25y\geq2000\Rightarrow1.25y\geq500\Rightarrow y\geq400 \). So \( (1500,400) \) (check: \( 1500 + 1.25\times400=1500 + 500=2000\geq2000 \))
• When \( x = 1000,y = 1000 \): \( 1000+1.25\times1000=1000 + 1250=2250\geq2000 \). So \( (1000,1000) \) is a solution.

Step2: Plot the Points

Plot \( (0,1600) \), \( (2000,0) \), \( (500,1200) \), \( (1000,800) \), \( (1500,400) \), \( (1000,1000) \) (or other valid points) on the shaded region from part (b).

Answer:

The inequality is \( \boldsymbol{x + 1.25y \geq 2000} \) (where \( x \) = number of hot dogs, \( y \) = number of sodas).

Part (b)