Question

Find the linear inequations for which the shaded area in Fig. 15.41 is the solution set. Draw the diagram of the solution set of the linear inequations:

Figure

Answer 1

Considering the line 2x + 3y = 6, we find that the shaded region and the origin (0, 0) are on the opposite side of this line and (0, 0) does not satisfy the inequation 2x + 3y $\ge$ 6 So, the first inequation is 2x + 3y $\ge$ 6

Considering the line 4x + 6y = 24, we find that the shaded region and the origin (0, 0) are on the same side of this line and (0, 0) satisfies the inequation 4x + 6y $\le$ 24 So, the corresponding inequation is 4x + 6y $\le$ 24

Considering the line x $-$ 2y = 2, we find that the shaded region and the origin (0, 0) are on the same side of this line and (0, 0) satisfies the inequation x $-$ 2y $\le$ 2 So, the corresponding inequation is x $-$ 2y $\le$ 2

Considering the line $-$3x + 2y = 3, we find that the shaded region and the origin (0, 0) are on the same side of this line and (0, 0) satisfies the inequation $-$3x + 2y $\le$ 3 So, the corresponding inequation is $-$3x + 2y $\le$ 3

Also, the shaded region is in the first quadrant. Therefore, we must have x $\ge 0\mathrm{and}y\ge 0$

Thus, the linear inequations comprising the given solution set are given below:
2x + 3y $\ge$ 6, 4x + 6y $\le$ 24, x $-$ 2y $\le$ 2, $-$3x + 2y $\le$ 3, x$\ge 0\mathrm{and}y\ge 0$