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

Verified step by step guidance

1. Step 1: Identify the inequalities in the system. The first inequality is $x^2+y^2\backslash leq\; 16$, which represents all points (x, y) inside or on the circle centered at the origin with radius 4, since \( \sqrt{16} = 4 \).
2. 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 $y\; =\; x^2\; -\; 4$.
3. 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.
4. 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.
5. 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.

Key Concepts

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.

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.

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.