Question

The shaded region in the figure is the solution set of the in equations.

A. $5x + 4y \geqslant 20$, $x \leqslant 6$,$y \geqslant 3$, $x \geqslant 0$,$y \geqslant 2$
B. $5x + 4y \geqslant 20$,$x \geqslant 6$, $y \leqslant 3$, $x \geqslant 0$,$y \geqslant 2$
C.$5x + 4y \geqslant 20$,$x \leqslant 6$, $y \leqslant 3$, $x \geqslant 0$,$y \geqslant 0$
D. $5x + 4y \leqslant 20$, $x \leqslant 6$, $y \leqslant 3$, $x \geqslant 0$,$y \geqslant 2$

Answer 1

First we will see from the given figure that the shaded region lies above $x$–axis, the shaded region is on the left of $x = 6$, the shaded region is below $y = 3$, lies above $y$–axis and then use the formula of the equation of a line is$y - {y_1} = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}\left( {x - {x_1}} \right)$, where line passing through the points $\left( {{x_1},{y_1}} \right)$ and $\left( {{x_2},{y_2}} \right)$ to find the required value.

Complete step-by-step answer:
Since we have seen in the given figure that the shaded region lies above $x$–axis, we get
$\Rightarrow y \geqslant 0$
Also from the given figure, the shaded region is on the left of $x = 6$, we get
$\Rightarrow x \leqslant 6$

So from the given figure, the shaded region is below $y = 3$, we get
$\Rightarrow y \leqslant 3$
Since we have seen in the given figure that the shaded region lies above $y$–axis, we get
$\Rightarrow x \geqslant 0$
We know that the formula of the equation of a line is$y - {y_1} = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}\left( {x - {x_1}} \right)$, where line passing through the points $\left( {{x_1},{y_1}} \right)$ and $\left( {{x_2},{y_2}} \right)$.
Substituting the value of points $\left( {4,0} \right)$ and $\left( {0,0} \right)$ the above formula of equation of line, we get

$$\Rightarrow y - 0 = \dfrac{{5 - 0}}{{0 - 4}}\left( {x - 4} \right) \\\ \Rightarrow y = - \dfrac{5}{4}\left( {x - 4} \right) \\\$$

Cross-multiplying the above equation, we get

$$\Rightarrow 4y = - 5\left( {x - 4} \right) \\\ \Rightarrow 4y = - 5x + 20 \\\$$

Adding the above equation with $5x$ on both sides, we get

$$\Rightarrow 4y + 5x = - 5x + 20 + 5x \\\ \Rightarrow 5x + 4y = 20 \\\$$

If we take $5x + 4y \geqslant 20$, put $\left( {0,0} \right)$ in the equation, we get

$$\Rightarrow 5\left( 0 \right) + 4\left( 0 \right) \geqslant 20 \\\ \Rightarrow 0 + 0 \geqslant 20 \\\ \Rightarrow 0 \geqslant 20 \\\$$

Since the above result is false, so $5x + 4y \leqslant 20$.
Thus, we have found out that $5x + 4y \geqslant 20$,$x \leqslant 6$, $y \leqslant 3$, $x \geqslant 0$,$y \geqslant 0$.
Hence, option C is correct.

Note: We need to know when the top side of the boundary line for the inequality symbols $>$ or $\geqslant$. And the bottom side of the boundary line for the inequality symbols $<$ or$\leqslant$. We have to be careful and examine the region carefully to avoid incorrect options. Since each of the options are really similar so mark the correct option carefully.