Chapter 5.7, Problem 46E

To determine

To graph: the system of linear inequalities that represents the situation.

Answer to Problem 46E

The inequalities that represent the given situation are $x\le 10\text{ and }y\le 2.$

Explanation of Solution

Given:

It can buy $9$ t-shirts and $1$ sweatshirt but it cannot buy $3$ t-shirts and $8$ sweatshirts.

Concept used:

Graphing the inequalities, it will graph the ordinary linear functions just like it is done before. The difference is that a solution to the inequality is not the drawn line but the area of the coordinate plane that satisfy the inequality. the boundary line is dashed for $<\text{or >}$ and solid line for $\le \text{or }\ge$ The common area is the solution of inequality.

There will be two signs for the domain inside the modulus.

If $\left|y\right|<x$ it implies:

  $y<x\text{ and }y>-x$ .

Calculation:

Write the linear inequalities for the solution where it can buy $9$ t-shirts and $1$ sweatshirt but it cannot buy $3$ t-shirts and $8$ sweatshirts.

Graph the linear inequalities that represents the given situation.

Consider the inequalities to describe the situation where vit can buy no more than $2$ sweatshirts and no more than $10$ t shirts.

Let $x$ be the number of t shirt and $y$ be the number of sweatshirts. Then the inequalities are:

  $\begin{array}{l}x\le 10.\\ y\le 2.\end{array}$

Hence, the inequalities that represent the given situation are:

  $x\le 10\text{ and }y\le 2.$

Graph the inequalities $x\le 10\text{ and }y\le 2$ as shown in the below figure. The intersection of blue and red colors represents the solution set of given system of inequalities. The ordered pair $\left(x,y\right)=\left(9,1\right)$ belongs to the solution set but the ordered pair $\left(x,y\right)=\left(3,8\right)$ does not belong to the solution set.

Graph of the inequalities $x\le 10\text{ and }y\le 2$ :

Hence, the graph of linear inequalities that represents the given situation is drawn.