Chapter 8.2, Problem 54MR

To determine

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

Answer to Problem 54MR

The solution of inequality $7x-2y\le 8$ for y is $y\ge 3.5x-4$ , and the graph of inequality $y\ge 3.5x-4$ is shown in Figure-1.

Explanation of Solution

Given information:

The given system of inequality is $7x-2y\le 8$

Concept used:

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

Calculations:

The given inequality is $7x-2y\le 8$ ,

  $\Rightarrow 7x-7x-2y\le 8-7x$  Subtracting 7x from each side $\Rightarrow \cancel{7x}\cancel{-7x}-2y\le 8-7x$

  $\Rightarrow -2y\le 8-7x$

  $\Rightarrow -2y\le 8-7x$   Multiplying each side by a -1 change the sign of inequality

  $\Rightarrow -2y\times -1\ge \left(8-7x\right)\times -1$

  $\Rightarrow 2y\ge 7x-8$

  $\Rightarrow \frac{2y}{2}\ge \frac{7x-8}{2}$   Dividing each side by 2

  $\Rightarrow y\ge 3.5x-4$

Thus, the solution of inequality $7x-2y\le 8$ for y is $y\ge 3.5x-4$ .

The graph of inequality $y\ge 3.5x-4$ is shown in Figure-1 here.

  

Interpretation: All the points of the shaded region are the solutions of inequality $y\ge 3.5x-4$ .

Conclusion:

The solution of inequality $7x-2y\le 8$ for y is $y\ge 3.5x-4$ , and the graph of inequality $y\ge 3.5x-4$ is shown in Figure-1.