Chapter ISG, Problem 6.6.6P

To determine

To write: and graph a system of inequalities that models this situation and identify the solution set.

Answer to Problem 6.6.6P

System of inequalities is $y\le 10-\frac{2}{3}x,x\ge 5,y\ge 3$

Explanation of Solution

Given information: L has $30 to purchase pens and pencils for her class. A three-pack

of pens costs $2, and a dozen pencils costs $3. Everyone needs at least one pen and two pencils. There are 15 students in her class including L.

Calculation:

Let L purchase x number of pens packets (each packet has three pens).

Let L purchase y number of pencils packets (each packet has a dozen (12) pencils).

Since, everyone needs at least one pen and two pencils and there are 15 students in her class including L.

So, $x\ge 5\text{and}y\ge 3$ because 15 students need at least 15 pens and 30 pencils so need at least 5 packets of pen and 3 packets of pencils (12.3=36).

Therefore $x\ge 5\text{and}y\ge 3$ .

Since L has $30 to purchase so system of inequalities in this situation will be:

  $\begin{array}{l}2x+3y\le 30(\text{cost}\text{of}\text{each}\text{pen}\text{packet}\text{is}\text{\$2,cost}\text{of}\text{}\text{each}\text{}\text{pencil}\text{}\text{packet}\text{is}\text{\$3)}\\ 3y\le 30-2x\\ y\le 10-\frac{2}{3}x\end{array}$

So system of inequalities is

  $y\le 10-\frac{2}{3}x,x\ge 5,y\ge 3$ .

The graph of the above inequality and its solution set is given below.

  

Solution set of the inequality is given by dark black area of the graph.