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Chapter 7: Problem 18

Graph inequality. \((x+2)^{2}+(y-1)^{2}<16\)

Short Answer

Expert verified
For the inequality \((x+2)^{2}+(y-1)^{2}<16\), the graph is the interior of the circle with a center at (-2,1) and a radius of 4 units.

Step by step solution

01

Identify Center and Radius from Given Equation.

Rewrite the given equation \((x+2)^{2}+(y-1)^{2}<16\) in the standard form. In this case, the center (h, k) is at (-2, 1), and the radius \(r\) is \(\sqrt{16} = 4\).
02

Graph the Circle.

Using the center and the radius identified in Step 1, draw a circle with the center at (-2, 1) point, and the radius equal to 4 units. Make sure to draw this circle with a dotted line, since the inequality is less than 16, not less than or equal to 16. It means that the points on the boundary circle aren't included in the solution.
03

Shade the Correct Area.

Since our inequality is less than 16, we will shade the region inside the circle. This shaded region that's inside represents all points (x, y) that are a distance less than 16 from the center (-2, 1).

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Key Concepts

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

Understanding the Center of a Circle
The center of a circle is a key concept when dealing with equations of circles, especially in inequalities. A circle is symmetrically fixed around its center point, which is denoted by the coordinates \((h, k)\). For the given inequality equation, \((x+2)^{2}+(y-1)^{2}<16\), the center can be found by comparing it to the general form \((x-h)^{2}+(y-k)^{2}=r^{2}\).

  • Here, we identify \(h\) as -2 and \(k\) as 1.
  • This tells us that the center of the circle is at the point \((-2, 1)\).
Knowing the center is crucial as it is the point from which every point on the boundary of the circle's radius is equidistant. In graphing, it determines where the circle is located on the plane.
The Role of the Radius in Graphing a Circle
The radius is the fixed distance from the center of the circle to any point on its circumference. In the context of graphing inequalities, the radius helps in determining the size of the circle.

For the inequality \((x+2)^{2}+(y-1)^{2}<16\), the term on the right-hand side under the square root gives the square of the radius \(r^{2}\). Thus, to find \(r\), we take the square root of 16, which is 4.

  • This means every point on the circle is precisely 4 units away from the center point \((-2, 1)\).
  • Graphically, it establishes the boundary within which all solutions for the inequality lie if it were the equation of a circle.
A circle drawn with this radius visually separates the region of interest, but remember, in inequalities, it can also indicate limits of inclusion based on whether the inequality involves "less than" or "less than or equal to" relations.
Shading in Inequalities: Identifying the Solution Space
Shading is a crucial step when graphing inequalities as it helps visualize the set of all points that satisfy the inequality. In this given problem, you have an inequality \((x+2)^{2}+(y-1)^{2}<16\), which implies that you are looking for the region where all points \((x,y)\) have a sum of squares less than 16.

  • Because the inequality symbol is "<", the circle itself, drawn as a boundary, is not included. We use a dotted line for the circumference.
  • Shade the entire area inside this circle, indicating all points that satisfy the inequality by being strictly within the radius from the center.
Shading beyond this dotted line circle would imply including points where the distance between the center and them is equal to or greater than the radius, which does not satisfy our inequality. Thus, shading correctly ensures visual clarity and accuracy in representing solutions.

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Most popular questions from this chapter

Explain how to solve a system of equations using the substitution method. Use \(y=3-3 x\) and \(3 x+4 y=6\) to illustrate your explanation.

Use a system of linear equations to solve Exercises. The current generation of college students grew up playing interactive online games, and many continue to play in college. The bar graph shows the percentage of U.S. college students playing online games, by gender. (GRAPH CAN'T COPY) A total of \(41 \%\) of college men play online games multiple times per day or once per day. The difference in the percentage who play multiple times per day and once per day is \(7 \% .\) Find the percentage of college men who play online games multiple times per day and the percentage of college men who play online games once per day.

What is a system of linear inequalities?

Write an equation involving \(a, b,\) and \(c\) based on the following description: When the value of \(x\) in \(y=a x^{2}+b x+c\) is \(4,\) the value of \(y\) is 1682.

Write a system of equations having \(\\{(-2,7)\\}\) as a solution set. (More than one system is possible.)
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