Question

1. colton is building a rectangular fence. he wants the length of the rectangle to be at least 25 yards and the perimeter to be no more than 150 yards. write and solve a system of linear inequalities that shows all possible combinations of the dimensions of the rectangle.
2. identify the variables:

x =
y =

1. write a system of inequalities:
2. graph the system:
3. list 2 possible combinations:

1. emma is buying new herbs for her garden. basil plants cost $5 each and mint plants cost $10 each. she wants to buy at least 9 plants but cannot spend more than $70. write and solve a system of linear inequalities that shows all possible combinations of the plants emma could purchase.
2. identify the variables:

x =
y =

1. write a system of inequalities:
2. graph the system:
3. list 2 possible combinations:

1. micah is having a party and is serving pizza and hamburgers. he wants enough for at least 20 guests. hamburgers are $4 each and pizza plates are $2 each. micah wants to spend no more than $80 on food for his party. write and solve a system of linear inequalities that shows all possible combinations of hamburgers and pizza plates he can buy.
2. identify the variables:

x =
y =

1. write a system of inequalities:
2. graph the system:
3. list 2 possible combinations:

Explanation:

Problem 3

Step1: Define variables

Let $x$ be the length of the rectangle and $y$ be the width of the rectangle.

Step2: Write inequalities

The length should be at least 25 yards, so $x\geq25$. The perimeter $P = 2(x + y)$ and it should be no more than 150 yards, so $2(x + y)\leq150$ which simplifies to $x + y\leq75$.

Step3: Find possible combinations

We can find combinations by considering the inequalities. For example, if $x = 25$, then from $x + y\leq75$, we have $y\leq50$. Two possible combinations are $(x = 25,y = 10)$ and $(x = 30,y = 45)$.

Step1: Define variables

Let $x$ be the number of basil plants and $y$ be the number of mint plants.

Step2: Write inequalities

She wants to buy at least 9 plants, so $x + y\geq9$. The cost of basil plants is $5x$ and of mint plants is $10y$, and she cannot spend more than $70$, so $5x+10y\leq70$ or $x + 2y\leq14$.

Step3: Find possible combinations

If $x=4,y = 5$, then $x + y=9$ and $5x+10y=5\times4 + 10\times5=20 + 50 = 70$. If $x=6,y = 3$, then $x + y=9$ and $5x+10y=5\times6+10\times3=30 + 30=60$.

Step1: Define variables

Let $x$ be the number of hamburgers and $y$ be the number of pizza plates.

Step2: Write inequalities

He wants enough for at least 20 guests, so $x + y\geq20$. The cost of hamburgers is $4x$ and of pizza plates is $2y$, and he wants to spend no more than $80$, so $4x+2y\leq80$ or $2x + y\leq40$.

Step3: Find possible combinations

If $x = 10,y = 10$, then $x + y=20$ and $4x+2y=4\times10+2\times10=40 + 20=60$. If $x = 15,y = 5$, then $x + y=20$ and $4x+2y=4\times15+2\times5=60 + 10 = 70$.

Answer:

1. $x$: length of the rectangle, $y$: width of the rectangle

2.

$$\begin{cases}x\geq25\\x + y\leq75\end{cases}$$

1. (Graphing requires drawing the lines $x = 25$ and $x + y=75$ and shading the appropriate region, which is difficult to show in text - but the region is to the right of $x = 25$ and below $x + y = 75$ in the first - quadrant)
2. $(25,10),(30,45)$

Problem 4