Q10. - Algebra 2

Question

For exercises 10 and 11, use the following information. 

Willis has been sent to the grocery store to purchase bagels and muffins for the members of the track team. He can spend at most \(28. A package of bagels costs \)2.50 and contains 6 bagels. A package of muffins costs $3.50 and contains 8 muffins. He needs to buy at least 12 bagels and 24 muffins.

10. Graph the region that shows how many packages of each item he can purchase. 

Step-by-Step Solution

Answer

The shaded region represents packages of bagels and muffins he can buy.

Step-1 – Define the variable

He has to purchase bagels and muffins.

He can spend at most $28. A package of bagels costs $2.50 and contains 6 bagels. 

A package of muffins costs $3.50 and contains 8 muffins. 

He needs to buy at least 12 bagels and 24 muffins.

Let the number bagels are x and number of muffins are y.

Step-2 – Construct system of inequalities

The system of inequalities that represent the data are provided below.

He needs to buy at least 12 bagels and 24 muffins.

$\begin{array}{c}x\ge 2\\ y\ge 3\end{array}$

He can spend at most $28. A package of bagels costs $2.50 and contains 6 bagels. 

A package of muffins costs $3.50 and contains 8 muffins. 

$2.50x+3.50y\le 28$

Therefore, the system of inequalities is provided below.

$\begin{array}{c}2.50x+3.50y\le 28\\ x\ge 2\\ y\ge 3\end{array}$

Step-3 – Apply the concept of graphing the inequality

The steps to graph the inequality are provided below.

1. If the inequality contains greater than or less than sign then the boundary of the line will be dashed. If the inequality contains signs of greater than or equal to or less than or equal to then the boundary of the line will be solid. 

2. Select a point (known as test point) from the plane that does not lie on the boundary on the line and substitute it in the inequality. 

3. If the inequality is true then shade the region that contains the test point otherwise shade the other region when inequality is false.

Step-4 – Interpret the sign of the inequality

Consider the inequality provided below.

$2.50x+3.50y\le 28$

The inequality contains the sign of less than or equal to.

Therefore, the boundary line will be solid.

Next, consider the inequalities.

$\begin{array}{c}x\ge 2\\ y\ge 3\end{array}$

The inequality contains the sign of greater than or equal to.

Therefore, the boundary line will be solid.

Step-5 – Graph the inequalities

Graph the inequalities $2.50x+3.50y\le 28$ and $x\ge 2\text{ and }y\ge 3$ on same plane and shade the region.

The corresponding equation is $2.50x+3.50y=28$.

Take a test point that does not lie on the boundary of the line, say $\left(0,0\right)$.

Substitute the point $\left(0,0\right)$ in the inequality and check whether it’s true or not.

$\begin{array}{c}2.50\left(0\right)+3.50\left(0\right)\le 28\\ 0\le 28\end{array}$

This is true.

Therefore, shade the region containing the point $\left(0,0\right)$.

Draw the line $x=2$.

Take a test point that does not lie on the boundary of the line, say $\left(0,0\right)$. 

Substitute the point $\left(0,0\right)$ in the inequality and check whether it’s true or not.

$0\ge 2$

This is false.

Therefore, shade the region not containing the point $\left(0,0\right)$.

Draw the line $y=3$.

Take a test point that does not lie on the boundary of the line, say $\left(0,0\right)$. 

Substitute the point $\left(0,0\right)$ in the inequality and check whether it’s true or not.

$0\ge 3$

This is false.

Therefore, shade the region not containing the point $\left(0,0\right)$.

Thus, the shaded regions are provided below.

Draw the line $x=2$.

Take a test point that does not lie on the boundary of the line, say $\left(0,0\right)$. 

Substitute the point $\left(0,0\right)$ in the inequality and check whether it’s true or not.

$0\ge 2$

This is false.

Therefore, shade the region not containing the point $\left(0,0\right)$.

The region 1 and region 4 corresponds to inequality $x\ge 2$.

The region 2 and region 4 corresponds to inequality $y\ge 3$.

The region 3 and region 4 corresponds to inequality $2.50x+3.50y\le 28$.

Step-6 – Shade the common region

The region common to both the inequalities $2.50x+3.50y\le 28$ and $x\ge 2\text{ and }y\ge 3$ is Region 4.

Hence, the shaded region represents packages of bagels and muffins he can buy.