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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 53
Chapter 6, Problem 53

In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. x2+y2≤1, y−x2>0
Graph showing the solution set for the inequalities x^2 + y^2 ≤ 16 and y - x^2 > -4.

Verified step by step guidance
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Step 1: Identify the inequalities in the system. The first inequality is x2 + y2 \(\leq\) 16, which represents all points (x, y) inside or on the circle centered at the origin with radius 4, since \( \sqrt{16} = 4 \).
Step 2: The second inequality is y - x^2 > -4. Rearranging this, we get y > x^2 - 4. This represents the region above the parabola y = x^2 - 4.
Step 3: To graph the solution set, first draw the circle x^2 + y^2 = 16. Shade the interior and the boundary because of the 'less than or equal to' sign.
Step 4: Next, graph the parabola y = x^2 - 4. Since the inequality is strict (greater than), shade the region above this parabola, not including the parabola itself.
Step 5: The solution set to the system is the intersection of the two shaded regions: points inside or on the circle and above the parabola. Identify this overlapping region on the graph as 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 Inequalities in Two Variables

Graphing inequalities involves shading regions of the coordinate plane that satisfy the inequality. For example, the inequality x² + y² ≤ 16 represents all points inside or on the circle centered at the origin with radius 4. Understanding how to graph such regions is essential for visualizing solution sets.
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Equations with Two Variables

Systems of Inequalities

A system of inequalities requires finding the intersection of solution sets for each inequality. The solution to the system is the region where all inequalities overlap. This concept is crucial for determining the combined feasible region that satisfies all given conditions.
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Quadratic Functions and Their Graphs

Quadratic functions like y = x² produce parabolas. Inequalities involving quadratics, such as y - x² > -4, describe regions relative to these parabolas. Recognizing the shape and position of these graphs helps in accurately shading the solution regions.
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Graphs of Logarithmic Functions
Related Practice
Textbook Question

Solve each system for x and y, expressing either value in terms of a or b, if necessary. Assume that a ≠ 0 and b ≠ 0. For the linear function f(x) = mx + b, f(−2) = 11 and ƒ(3) = -9. Find m and b.

Textbook Question

Graph the solution set of each system of inequalities or indicate that the system has no solution.

{(x−1)2+(y+1)2<25(x−1)2+(y+1)2≥16\(\begin{cases}\)(x - 1)^2 + (y + 1)^2 < 25 \\(x - 1)^2 + (y + 1)^2 \(\geq\) 16\(\end{cases}\)

Textbook Question

In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. x2+y2<16, y≥2x

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Textbook Question

In Exercises 47–52, solve each system by the method of your choice. {3x2+1y2=75x2−2y2=−3\(\begin{cases}\]\frac{3}{x^2}\) + \(\frac{1}{y^2}\) = 7 \(\frac{5}{x^2}\) - \(\frac{2}{y^2}\) = -3\(\end{cases}\)

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Textbook Question

Find the value of the objective function z = 2x + 3y at each corner of the graphed region shown. What is the maximum value of the objective function? What is the minimum value of the objective function?

Textbook Question

Find the length and width of a rectangle whose perimeter is 36 feet and whose area is 77 square feet.