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Chapter 8: Problem 21

Sketch the graph of each nonlinear inequality. $$x^{2}+y^{2} \geq 1$$

Short Answer

Expert verified
Sketch a circle centered at (0,0) with radius 1, and shade everything outside the circle and including the circle boundary itself.

Step by step solution

01

Recognize the inequality as a circle

The inequality given, \(x^{2}+y^{2} \geq 1\), resembles the equation of a circle, \(x^2 + y^2 = r^2\), where \r = 1\. This represents a circle centered at (0,0) with a radius of 1.
02

Draw the circle

Sketch a circle centered at the origin (0,0) with a radius of 1. This is done by plotting points that form the circle, such as (1,0), (-1,0), (0,1), and (0,-1). Then connect these points smoothly.
03

Determine the region defined by the inequality

The inequality \(x^{2}+y^{2} \geq 1\) implies we include the area outside the circle and the points on the circle boundary itself since \geq\ includes equality. Shade the region outside the circle.
04

Verify the shade region with a test point

Choose a point not on the circle to confirm it satisfies the inequality. For instance, (2,0) gives \(2^2 + 0^2 = 4 \geq 1\), which is true. Therefore, the shaded region outside the circle is correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

circle equation
When dealing with nonlinear inequalities, it's essential to recognize familiar equations. The given inequality, \(x^{2} + y^{2} \geq 1\), looks similar to the equation of a circle. A circle's equation follows the form \(x^2 + y^2 = r^2\), where \(r\) is the radius. In this case, our radius is 1, since our equation matches \(x^2 + y^2 = 1\). This means the circle is centered at the origin (0,0), and every point on this circle is exactly 1 unit away from the center.
inequality regions
Inequalities tell us about regions rather than exact lines or curves. The inequality given is not just about the boundary of the circle but also about the area outside or inside. Since our inequality is \(x^{2} + y^{2} \geq 1\), this means all the points forming the circle and those outside of it satisfy the condition. Always pay attention to the inequality sign. If it were \(x^{2} + y^{2} < 1\), it would include just the area inside the circle, not on it.
graph sketching
To visually represent these mathematical concepts, we need to sketch the graph. Start by drawing a circle centered at (0,0) with the correct radius. Plot points such as (1,0), (-1,0), (0,1), and (0,-1), then connect the dots smoothly to form a circle. Next, since our inequality is \(\geq\), shade the area outside the circle. This shading implies all points from the boundary outwards meet our inequality's requirement.
test points
Using test points ensures the shaded region is correct. Choose a point in the shaded area, such as (2,0). Substitute these coordinates into the inequality: \(2^2 + 0^2 = 4\). Since 4 is greater than 1, the point satisfies the inequality. If the point were inside the circle, like (0.5,0.5), its coordinates would give \(0.5^2 + 0.5^2 = 0.5\), which doesn't satisfy \(\geq 1\). Therefore, confirm the shaded area includes and correctly represents all points meeting the inequality.

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Most popular questions from this chapter

Solve each system. $$\begin{aligned} x+y+z &=9000 \\ 0.05 x+0.06 y+0.09 z &=710 \\ z &=3 y \end{aligned}$$

Solve each system. $$\begin{aligned} x+y+2 z &=7.5 \\ 3 x+4 y+z &=12 \\ 5 x+2 y+5 z &=21 \end{aligned}$$

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