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Ch. 5 - Systems of Equations and Inequalities
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 5 - Systems of Equations and InequalitiesProblem 17
Chapter 6, Problem 17

Graph each inequality. (x−2)2+(y+1)2<9

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Identify the inequality as representing a circle in the coordinate plane. The general form of a circle's equation is \( (x - h)^2 + (y - k)^2 = r^2 \), where \((h, k)\) is the center and \(r\) is the radius.
Compare the given inequality \( (x - 2)^2 + (y + 1)^2 < 9 \) to the general form. Here, the center is at \( (2, -1) \) and the radius \( r \) is \( \sqrt{9} = 3 \).
Since the inequality is '<' (less than), the graph represents all points inside the circle but not on the boundary. This means the circle itself is not included, so the boundary should be drawn as a dashed curve.
Plot the center point at \( (2, -1) \) on the coordinate plane. Then, using the radius 3, mark points 3 units away from the center in all directions (up, down, left, right) to help sketch the circle.
Shade the interior region of the circle to represent all points \( (x, y) \) that satisfy the inequality \( (x - 2)^2 + (y + 1)^2 < 9 \). This shaded area shows the solution set.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Graphing Circles

The inequality (x−2)² + (y+1)² < 9 represents all points inside a circle centered at (2, -1) with radius 3. Understanding how to graph circles involves identifying the center and radius from the equation and plotting the circle accordingly.
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Circles in Standard Form

Inequalities and Regions

An inequality like (x−2)² + (y+1)² < 9 describes a region inside the circle, not just the boundary. Points satisfying the inequality lie strictly inside the circle, so the graph includes all interior points but excludes the circle itself.
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Systems of Inequalities

Coordinate Plane and Plotting Points

Plotting the graph requires familiarity with the coordinate plane, including locating points using (x, y) coordinates. This skill helps in accurately drawing the circle's center and radius, and shading the correct region defined by the inequality.
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Graphs & the Rectangular Coordinate System
Related Practice
Textbook Question

Solve each system in Exercises 5–18. {3(2x+y)+5z=−12(x−3y+4z)=−94(1+x)=−3(z−3y)\(\begin{cases}\)3(2x + y) + 5z = -1 \\2(x - 3y + 4z) = -9 \\4(1 + x) = -3(z - 3y)\(\end{cases}\)

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Textbook Question

Write the partial fraction decomposition of each rational expression. x/(x2 +2x -3)

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In Exercises 16–24, write the partial fraction decomposition of each rational expression. x/(x - 3)(x + 2)

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In Exercises 5–18, solve each system by the substitution method. y = (1/3)x + 2/3 y = (5/7)x - 2

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Textbook Question

In Exercises 1–18, solve each system by the substitution method. {x+y=1(x−1)2+(y+2)2=10\(\begin{cases}\)x + y = 1 \\(x - 1)^2 + (y + 2)^2 = 10\(\end{cases}\)