The shaded region in the following figure is the solution set of the inequations

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From the given graph, we can identify the boundary lines of the shaded region.

The line passing through $(0, 5)$ and $(3, 0)$ has the equation $\frac{x}{3} + \frac{y}{5} = 1 \Rightarrow 5 x + 3 y = 15$ . The shaded region lies above this line (away from the origin), so the inequality is $5 x + 3 y \geq 15$ .

The line passing through $(0, 3)$ and $(6, 0)$ has the equation $\frac{x}{6} + \frac{y}{3} = 1 \Rightarrow x + 2 y = 6$ . The shaded region lies above this line, so the inequality is $x + 2 y \geq 6$ .

The vertical boundary line is at $x = 7$ , and the shaded region is to its left, giving $x \leq 7$ .

The horizontal boundary line is at $y = 6$ , and the shaded region is below it, giving $y \leq 6$ .

Since the region is in the first quadrant, we also have $x \geq 0$ and $y \geq 0$ .

Combining all these, the system of inequalities is $x + 2 y \geq 6$ , $5 x + 3 y \geq 15$ , $x \leq 7$ , $y \leq 6$ , $x, y \geq 0$ .

Answer: $x + 2 y \geq 6, 5 x + 3 y \geq 15, x \leq 7, y \leq 6, x y \geq 0$

Mathematics - Area Under Curves Question with Solution | TestHub

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