Question
Find the linear inequations for which the shaded area in Fig. is the solution set. Draw the diagram of the solution set of the linear inequations:
✓
Answer
Consider the line 2x + 3y = 6. we observe that the shaded region and the origin are on the opposite sides of the line 2x + 3y = 6 and (0, 0) does not satisfy the inequation 2x + 3y 2 6. So, we must have one inequations as $2\text{x} + 3\text{y}\geq 26$
Consider the line 4x + 6y = 24. we observe that the shaded region and the origin are on the same side of the line 4x + 6y = 24 and (0, 0) satisfies the linear inequation $4\text{x} + 6\text{y}\leq 24.$
So, the second inequations is $4\text{x} + 6\text{y}\leq 24.$
Consider the line - 3x + 2y = 3. We observe that the shaded region and the origin are on the same side of the line - 3x + 2y = 3 and (0, 0) satisfies the linear inequation $-3\text{x}+2\text{y}\leq3.$ so, the third inequations is $-3\text{x}+2\text{y}\leq3.$
Finaly,consider the line x - 2y = 2. We observe that the shaded region and the origin are on the same side of the line x - 2y = 2 and (0, 0) satisfies the linear inequation $\text{x} - 2\text{y}\leq2.$ so, the forth inequations is $\text{x} - 2\text{y}\leq2.$
We also notice that the shaded region is above x-axis and is on the right side of y-axis. so, we must have $\text{x}\geq0$ and $\text{y}\geq0$
Thus, the linear inequations corresponding to the given solution set are
$2\text{x}+3\text{y}\geq6,4\text{x}+6\text{y}\leq24,-3\text{x}+2\text{y}\leq3,\text{x}-2\text{y}\leq2,\text{x}\geq0,\text{y}\geq0.$
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