Question 8.3.7: Find a system of inequalities for the shaded region shown in......

Precalculus Functions and Graphs [1893897]

Find a system of inequalities for the shaded region shown in Figure 10.

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Step 1:

The equation of the circle is given as x²+y²=5², which represents a circle with radius 5 centered at the origin (0,0).

Step 2:

Since the interior of the solid circle is shaded, we want to describe the region that includes the circle. This can be done using the inequality x²+y² ≤ 25, which represents all the points inside or on the boundary of the circle.

Step 3:

The exterior of the circle is represented by points that are not inside the circle. This can be described using the inequality x²+y²>25, which represents all the points outside the circle.

Step 4:

The shaded region is below the dashed line with equation y=3/4 x. To describe this region, we use the inequality y<3/4 x, which represents all the points below the line.

Step 5:

Lastly, the shaded region is above the solid horizontal line y=-3. To describe this region, we use the inequality y ≥ -3, which represents all the points above or on the line.

Step 6:

Combining all the inequalities, we have the system of inequalities: x²+y² ≤ 25 y < 3/4 x y ≥ -3

Final Answer

An equation of the circle is x²+y²=5². Since the interior of the solid circle is shaded, the shaded region (including the circle) can be described by x²+y² ≤ 25. The exterior of the circle could be described by x²+y²>25.

Because the shaded region is below the dashed line with equation $y=\frac{3}{4} x$ , it is described by the inequality $y<\frac{3}{4} x$ . Lastly, since the shaded region is above the solid horizontal line y=-3, we use y ≥-3. Thus, a system is