Chapter 8.9, Problem 48E

To determine

Whichinequality has no solutionsin the first quadrant?

Answer to Problem 48E

The inequality $-3x-y>6$ has no solutions in the first quadrant and option I is the correct choice.

Explanation of Solution

Given:

Four choices:

  $\text{F}\text{. }x<1\text{ G}\text{. }y\le -2x+6\text{ H}\text{. }x+y\le 5\text{ I}\text{. }-3x-y>6$

Calculation:

To answer our given problem, we will take one point from 1^st quadrant and substitute it in our given inequalities one by one.

We will check point $\left(\frac{1}{2},1\right)$ in our 1^st inequality.

  $\begin{array}{l}x<1\left(\text{Given inequality}\right)\\ \frac{1}{2}<1\left(\text{Substituting }x=\frac{1}{2}\right)\end{array}$

Since thepoint $\left(\frac{1}{2},1\right)$ satisfies our given inequality, so the inequality $x<1$ has a solution in 1^st quadrant.

We will check point $\left(1,1\right)$ in our 2^ndinequality.

  $\begin{array}{l}y\le -2x+6\left(\text{Given inequality}\right)\\ 1\le -2(1)+6\left(\text{Substituting }x=1\text{and }y=1\right)\\ 1\le -2+6\\ 1\le 4\end{array}$

Since the point $\left(1,1\right)$ satisfies our given inequality, so the inequality $y\le -2x+6$ has a solution in 1^st quadrant.

We will check point $\left(1,1\right)$ in our 3^rd inequality.

  $\begin{array}{l}x+y\le 5\left(\text{Given inequality}\right)\\ 1+1\le 5\left(\text{Substituting }x=1\text{and }y=1\right)\\ 2\le 5\end{array}$

Since the point $\left(1,1\right)$ satisfies our given inequality, so the inequality $x+y\le 5$ has a solution in 1^st quadrant.

Upon looking at our 4th inequality $-3x-y>6$ , we can see that no positive integers are solution for our given inequality. Since the coefficient of x and y terms are negative, so any positive integer will result in a negative number on left side of inequality.

There is a positive number on right side of inequality. Since all negative numbers are less than 6, therefore, the inequality $-3x-y>6$ has no solution in 1^st quadrant and option I is the correct choice.