Question

linear inequalities - real world situations
#1 madison is making cookies to sell for a mission trip. she makes $6 on each batch of snickerdoodles, x, and $4 on each batch of chocolate chip, y. she wants to make at least $1200 for her trip. write an inequality to represent this situation.
#2 graph the situation in #1.
#3 choose three points that are solutions for #1 and explain the meaning in the context of the situation.

Explanation:

Step1: Identify earnings from each type of cookie

Earnings from snickerdoodles is $6x$ (where $x$ is number of batches) and from chocolate - chip is $4y$ (where $y$ is number of batches). She wants at least $1200$, so the inequality is $6x + 4y\geq1200$.

Step2: Simplify the inequality

Divide the entire inequality $6x + 4y\geq1200$ by 2 to get $3x + 2y\geq600$.

Step3: Graph the inequality

First, rewrite the inequality as an equation $3x + 2y=600$. When $x = 0$, $y = 300$ (so the $y$-intercept is $(0,300)$). When $y = 0$, $x = 200$ (so the $x$-intercept is $(200,0)$). Plot these two points and draw a solid line (since the inequality is $\geq$). Then, test a point not on the line, say $(0,0)$. Substituting into $3x + 2y\geq600$ gives $0\geq600$ which is false. So, shade the region that does not contain the origin.

Step4: Find solution points

Point 1: $(200,0)$

This means Madison makes 200 batches of snickerdoodles and 0 batches of chocolate - chip cookies, and she makes $6\times200+4\times0 = 1200$ dollars.
Point 2: $(0,300)$

This means Madison makes 0 batches of snickerdoodles and 300 batches of chocolate - chip cookies, and she makes $6\times0 + 4\times300=1200$ dollars.
Point 3: $(100,150)$

She makes 100 batches of snickerdoodles and 150 batches of chocolate - chip cookies. Earnings are $6\times100+4\times150=600 + 600=1200$ dollars.

Answer:

#1: $6x + 4y\geq1200$ (or $3x + 2y\geq600$)
#2: Graph with $x$-intercept at $(200,0)$, $y$-intercept at $(0,300)$, solid line and shading above the line.
#3:

• $(200,0)$: 200 batches of snickerdoodles, 0 batches of chocolate - chip cookies, earnings of $1200$ dollars.
• $(0,300)$: 0 batches of snickerdoodles, 300 batches of chocolate - chip cookies, earnings of $1200$ dollars.
• $(100,150)$: 100 batches of snickerdoodles, 150 batches of chocolate - chip cookies, earnings of $1200$ dollars.