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Chapter 5: Problem 14

Graph each inequality. $$ x^{2}+y^{2} \leq 4 $$

Short Answer

Expert verified
The graph of the inequality is a filled circle, centered at the origin (0,0) with a radius of 2. This includes all points within the circle as well as the circle boundary.

Step by step solution

01

Identify the Circle

To identify the circle, consider the inequality as if it were an equation \( x^{2} + y^{2} = 4 \). Here, the equation represents a circle centered at the origin with a radius of 2 (since the radius \( r \) is the square root of 4).
02

Graph the Circle

Plot this circle by marking out the radius on both x and y axes. Draw the curve to complete the circle. Remember that since it's an inequality, we need to include the circle's interior as well. This can be shown by shading the interior of the circle, as that represents all the points which satisfy the inequality.
03

Confirm Solutions

Pick a test point inside the shaded circle (e.g. origin (0,0)) and substitute these values into the inequality to confirm if it's true. If the point satisfies the inequality, then the shading is correct. If not, the other side of the line should have been shaded.

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Key Concepts

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

Circle Graph
A circle graph is a way of representing data or a functioning area in geometry using a circular shape. This is particularly relevant when dealing with equations of the form \(x^2 + y^2 = r^2\), where \(r\) is the radius of the circle. The center of the circle given in standard form is at the origin (0,0).

When graphing a circle, you can follow these simple steps:
  • Identify the center and the radius from the equation. For \(x^2 + y^2 = 4\), the center is (0,0) and the radius is 2.
  • Plot the center point on the coordinate plane.
  • From the center, move outward in all directions the distance equal to the radius and mark these points.
  • Connect these points smoothly to form the circle.

This graphical representation allows one to visually inspect areas and can also be shaded or marked to denote inequalities where the area within or beyond represents solutions to an inequality.
Inequality Graphing
Inequality graphing involves shading regions on a graph to indicate a range of solutions, not just exact values. When dealing with inequalities in the form of a circle, like \(x^2 + y^2 \leq 4\), you need to decide if you should include the interior or exterior part of the circle.

Here's how to interpret and graph inequalities:
  • If the inequality symbol is \(\leq\) or \(\geq\), the boundary (circle) itself is included in the solution set.
  • If the inequality symbol is \(<\) or \(>\), the boundary is not included, typically represented by a dashed line.
  • Shade the region that satisfies the inequality. For \(x^2 + y^2 \leq 4\), shade the interior of the circle because it includes points with values less than or equal to 4.

These graphs are useful for visually solving problems and understanding which values make the inequality true. Always check by using test points to ensure the correct area is shaded.
Coordinate Geometry
Coordinate geometry, or analytic geometry, is the study of geometric figures using a coordinate system. By representing geometric shapes algebraically, you can solve complex problems that may involve lines, curves, and circles.

Fundamental concepts in coordinate geometry include:
  • The Cartesian plane: where two perpendicular axes, the x-axis and y-axis, intersect at the origin (0,0).
  • Points are located using ordered pairs (x, y).
  • Equations represent geometric figures, such as the circle \(x^2 + y^2 = r^2\), where \(r\) is the radius.
  • Inequalities define regions, such as \(x^2 + y^2 \leq r^2\), representing the area within the circle.

This field allows for a direct and visual approach to understanding geometry, making it easier to tackle problems that can be complex in pure algebraic terms.

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Most popular questions from this chapter

Explain how to graph \(2 x-3 y<6\).

In Exercises \(31-42,\) solve by the method of your choice. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets. \(\frac{x}{6}-\frac{y}{2}=\frac{1}{3}\) \(x+2 y=-3\)

Sketch the graph of the solution set for the following system of inequalities: $$ \begin{array}{r}|x+y| \leq 3 \\\|y| \leq 2\end{array} $$

In Exercises \(19-30,\) solve each system by the addition method. \(3 x=4 y+1\) \(3 y=1-4 x\)

In Exercises \(1-4,\) determine whether the given ordered pair is a solution of the system. $$ \begin{aligned} &(-3,5)\\\ &9 x+7 y=8\\\ &8 x-9 y=-69 \end{aligned} $$
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