Chapter 10.9, Problem 52E

To determine

The system of inequalities for number of table and chairs that the man and his daughter make daily and sketch the graph for solution set.

Answer to Problem 52E

The system of inequalities that describes the number of tables and chairs produced are $\{\begin{array}{l}3y+2x\le 12\\ y+2x\le 8\\ x\ge 0,y\ge 0\end{array}$, and the graph is shown in Figure (1).

Explanation of Solution

Given:

Let the number of unfinished chairs be x.

Let the number of unfinished tables be y.

The total number of hours required for sawing is 12

The total number of hours required for sawing of table is 3

The total number of hours required for sawing of chairs is 2

The first inequality is $2x+3y\le 12$.

The total number of hours required for assembly is 8

The total number of hours required for assembly of table is 1

The total number of hours required for assembly of chair is 2

The second inequality is $y+2x\le 8$

The number of chairs and tables cannot be negative. So

$x\ge 0$

$y\ge 0$

Take the first inequality $2x+3y\le 12$.

To plot the graph change the sign of inequality as $2x+3y=12$.

Substitute 0 for x in $2x+3y=12$

$y=4$.

Substitute 0 for y in $2x+3y=12$

$x=6$.

Take the second inequality $y+2x\le 8$.

To plot the graph change the sign of inequality as $y+2x=8$.

Substitute 0 for x in $y+2x=8$

$y=8$

Substitute 0 for y in $y+2x=8$

$x=4$

Now, $x\ge 0$ and $y\ge 0$ represents first quadrant.

The point of intersection between the inequality $3y+2x=12$ and $y+2x=8$.

$\begin{array}{l}2x+3y=12\\ \underset{\_}{2x+y=8}\left[\text{subtract}\right]\\ 2y=4\\ y=2\end{array}$

Substitute 2 for y in $2x+3y=12$.

$\begin{array}{c}2x+6=12\\ 2x=6\\ x=3\end{array}$

The intersection between $3y+2x=12$ and $y+2x=8$ is $\left(3,2\right)$.

Figure (1)

Thus, system of inequalities that describes the number of tables and chairs produced are $\{\begin{array}{l}3y+2x\le 12\\ y+2x\le 8\\ x\ge 0,y\ge 0\end{array}$, and the graph is shown in Figure (1).