Problem 54 - College Algebra Essentials

Question

Graph the solution set of each system of inequalities or indicate that the system has no solution. $$\left\\{\begin{aligned}x^{2}+y^{2} &<4 \\\y-x^{2} & \geq 0\end{aligned}\right.$$

Step-by-Step Solution

Answer

The solution is the internal region shared by the inequalities, including the points on the parabola \(y = x^{2}\), but excluding the points on the circle \(x^{2} + y^{2} = 4\).

Step 1: Understand the Components of the System

Before solving the system of inequalities, it's essential to understand the individual inequalities. The first inequality \(x^{2}+y^{2}<4\) represents the inner region of a circle with radius 2 centered at the origin. That is, it includes all the points that lie within this circle but excluding its boundary. The second inequality \(y-x^{2} \geq 0\) is represented by the region above the parabola \(y = x^{2}\), including the parabola itself.

Step 2: Graph the System of Inequalities

Graph both inequalities on the same coordinate grid. Use dashed lines to represent excluded boundaries.

Step 3: Identify and Highlight the Solution Set

This is the region where both of the inequalities intersect. Remember, because of the '>' sign in the second inequality, the boundary that belongs to the parabola is included in our solution.

Step 4: Confirm by Taking a Sample Point

To ensure our solution is correct, it's good practice to take a point from the identified region and verify that it satisfies both inequalities. If it does, then we can be sure the solution is correct.

Key Concepts

Graphing Inequalities

Graphing inequalities involves depicting regions on a coordinate plane that represent solutions to inequality equations. Unlike equations where solutions are specific points, inequalities encompass areas. To graph an inequality:

• First, identify the boundary line or curve. This could be a line, parabola, or circle.
• Determine if the boundary is included in the solution (solid line) or not (dashed line).
• Shade the region that satisfies the inequality. For 'less than' or 'greater than' (<, >), use a dashed boundary. For 'less than or equal to' or 'greater than or equal to' (≤, ≥), use a solid boundary.

To check correctness, choose a test point from the shaded region to verify if it satisfies the inequality.
This visual representation helps in understanding where the solutions lie, offering a clear view of feasible solutions.

Circle Equation

The equation of a circle in a coordinate plane is of the form \(x^2 + y^2 = r^2\). Here, \((x, y)\) are the coordinates, and \(r\) is the radius. The center of a circle at the origin has the form \(x^2 + y^2 = r^2\). If the inequality is \(x^2 + y^2 < r^2\), we are looking at the interior of the circle, not including the boundary.
Saying \(x^2 + y^2 < 4\) means the region inside a circle centered at (0, 0) with radius 2.

• Points such as (0,0) or (1,1) lie inside this region.
• Visualize by plotting with a circle but with a dashed boundary, as the line \(x^2 + y^2 = 4\) itself isn't included.

Understanding this helps in identifying how such a circle's equation behaves when depicted graphically.

Parabola Equation

Parabolas are represented by equations like \(y = ax^2 + bx + c\). In the case of \(y = x^2\), this is a simple upward-opening parabola with its vertex at the origin, namely (0,0).
The inequality \(y \geq x^2\) includes points on and above this parabola. So, when graphing:

• Draw the parabola using a solid line, as the boundary is included.
• The region above the curve is shaded, representing solutions to the inequality.

By understanding the nature of parabolas, you can accurately determine which parts of the graph represent solutions and how to shade them properly.

Solution Set Intersection

The intersection of solution sets involves finding the common region that satisfies all inequalities in a system. For systems like the one with a circle and parabola:

• Graph each inequality separately first.
• Use different shading patterns or colors to distinguish.
• The solution set is where these shaded regions overlap.

In our example, overlap represents points both inside the circle and on or above the parabola. This intersected region is the actual solution to the system. It’s visually confirmed on a graph and can be analytically checked by selecting confined-region test points.
Understanding solution set intersections is key to solving and visualizing systems of inequalities effectively.