Q. 4.118 - Intermediate Algebra

Question

Mary needs to purchase supplies of answer sheets and pencils for a standardized test to be given to the juniors at her high school. The number of the answer sheets needed is at least $5$ more than the number of pencils. The

pencils cost $\backslash (2$ and the answer sheets cost $\backslash )1$. Mary’s budget for these supplies allows for a maximum cost of

$\$400$.

ⓐ Write a system of inequalities to model this situation.

ⓑ Graph the system.

ⓒ Could Mary purchase $100$ pencils and $100$ answer sheets?

ⓓ Could Mary purchase $150$ pencils and $150$ answer sheets?

Step-by-Step Solution

Answer

part(a). The system of inequalities is:

$\left\{\begin{array}{l}y\le x+5\\ 2x+y\le 400\\ x\ge 0\\ y\ge 0\end{array}\right.$

Part (b). The graph of the system of inequalities:

part (c). Yes

Part(d). No

part (a) Step 1. Form the inequalities

Let

$$\begin{gathered} y=number of answer sheets \\ x=number of pencils \end{gathered}$$

The number of answer sheets is at least $5$ more than that of pencils. so,

 $y\le x+5$

$\$2$ for pencils and $\$1$ for answer sheet must not be greater than $\$400$. So,

$2x+y\le 400$

The number of pencils must be greater or equal to zero. so,

$x\ge 0$

The number of answer sheets must be greater or equal to zero. so, 

$y\ge 0$

Thus, The system of inequalities is: 

$\left\{\begin{array}{l}y\le x+5\\ 2x+y\le 400\\ x\ge 0\\ y\ge 0\end{array}\right.$

part (b) Step 1. graph the system of inequalities

The graph for the obtained system of inequalities is:

part( c) Step 1 . Find 100 , 100 in the solution region

To determine if $100$ pencils and $100$ answer sheets can be purchased.

we look at the graph to see if the point $\left(100,100\right)$ is in the solution region (dark blue).

Yes, the point $\left(100,100\right)$ is in the solution region. So, Mary can purchase $100$ pencils and $100$ answer sheets.

part (d) step 4. Find 150 , 150 in the solution region.

To determine if $150$ pencils and $150$ answer sheets can be purchased.

we look at the graph to see if the point $\left(150,150\right)$ is in the solution region (dark blue).

No, the point $\left(150,150\right)$ is not in the solution region. So, Mary cannot purchase $150$ pencils and $150$ answer sheets.