Chapter 8.2, Problem 55MR

To determine

Tosolve: The given inequality for $y$ and then graph the inequality.

Answer to Problem 55MR

The solution of inequality $-3y-x>9$ for y is $y<-\left(3+\frac{x}{3}\right)$ , and the graph of inequality $y<-\left(3+\frac{x}{3}\right)$ is shown in Figure-1.

Explanation of Solution

Given information:

The given system of inequality is $-3y-x>9$ .

Concept used:

Adding, subtracting, multiplying and dividing by a positive quantity on each side does not change the sign of an inequality.

Calculations:

The given inequality is $-3y-x>9$ ,

  $\Rightarrow -\left(3y+x\right)\times -1<9\times -1$  Multiplying each side by a -1 change the sign of inequality

  $\Rightarrow \left(3y+x\right)<-9$

  $\Rightarrow \left(3y+x\right)-x<-9-x$  Subtracting x from each side

  $\Rightarrow 3y+\cancel{x}-\cancel{x}<-\left(9+x\right)$

  $\Rightarrow 3y<-\left(9+x\right)$

  $\Rightarrow \frac{3y}{3}<-\frac{\left(9+x\right)}{3}$  Dividing each side by 3

  $\Rightarrow y<-\frac{\left(9+x\right)}{3}$

Thus, the solution of inequality $-3y-x>9$ for y is $y<-\frac{\left(9+x\right)}{3}=-\left(3+\frac{x}{3}\right)$ .

The graph of inequality $y<-\left(3+\frac{x}{3}\right)$ is shown in Figure-1 here.

  

Interpretation: All the points of the shaded region are the solutions of inequality $y<-\left(3+\frac{x}{3}\right)$ .

Conclusion:

The solution of inequality $-3y-x<9$ for y is $y<-\left(3+\frac{x}{3}\right)$ , and the graph of inequality $y<-\left(3+\frac{x}{3}\right)$ is shown in Figure-1.