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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 47
Chapter 6, Problem 47

Graph the solution set of each system of inequalities or indicate that the system has no solution.
{x2+y2≤16x+y>2\(\begin{cases}\)x^2 + y^2 \(\leq\) 16 \(\x\) + y > 2\(\end{cases}\)

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Identify the first inequality: \(x^{2} + y^{2} \leq 36\). This represents all points inside or on the circle centered at the origin \((0,0)\) with radius \(6\), since \(\sqrt{36} = 6\).
Identify the second inequality: \(3x + y > 6\). This is a linear inequality representing all points above the line \(3x + y = 6\).
Rewrite the linear inequality in slope-intercept form to better understand the boundary line: \(y > -3x + 6\). This line has a slope of \(-3\) and a y-intercept at \((0,6)\).
Graph the circle \(x^{2} + y^{2} = 36\) as a solid boundary because the inequality includes equality (\(\leq\)), meaning points on the circle are included.
Graph the line \(y = -3x + 6\) as a dashed line because the inequality is strict (\(>\)), meaning points on the line are not included. Then shade the region above this line where \(y > -3x + 6\).
The solution set to the system is the intersection of the shaded region inside the circle and above the line. This means you look for points that satisfy both inequalities simultaneously.

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Key Concepts

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

Graphing Circles and Inequalities

The inequality x² + y² ≤ 36 represents all points inside or on the circle centered at the origin with radius 6. Understanding how to graph this region involves recognizing the circle's boundary (x² + y² = 36) and shading the interior where the inequality holds true.
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Circles in Standard Form

Graphing Linear Inequalities

The inequality 3x + y > 6 represents a half-plane above the line 3x + y = 6. To graph it, first plot the boundary line, then determine which side satisfies the inequality by testing a point, shading the region where the inequality is true.
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Linear Inequalities

Solution Set of a System of Inequalities

The solution set of the system is the intersection of the regions satisfying each inequality. It includes points inside the circle and above the line simultaneously. Identifying this overlap is key to correctly graphing the system and understanding if solutions exist.
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Systems of Inequalities
Related Practice
Textbook Question

In Exercises 47–52, solve each system by the method of your choice. {2x2+xy=6x2+2xy=0\(\begin{cases}\)2x^2 + xy = 6 \(\x\)^2 + 2xy = 0\(\end{cases}\)

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Textbook Question

In Exercises 39–45, graph each inequality. y ≤ 2^x

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Textbook Question

In Exercises 47–48, solve each system by the method of your choice. (x + 2)/2 - (y + 4)/3 = 3 (x + y)/5 = (x - y)/2 - 5/2

Textbook Question

In Exercises 46–55, graph the solution set of each system of inequalities or indicate that the system has no solution.


This

is a piecewise function. Refer to the textbook.

Textbook Question

Perform each long division and write the partial fraction decomposition of the remainder term. (x4-x2+2)/(x3-x2)

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Textbook Question

In Exercises 46–55, graph the solution set of each system of inequalities or indicate that the system has no solution.


This

is a piecewise function. Refer to the textbook.