Problem 54 - College Algebra

Question

In Exercises 27–62, 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 region that lies inside the circle \(x^{2}+y^{2} < 4\) but above the parabola \(y \geq x^{2}\).

Step 1: Graph the first inequality

The first inequality \(x^{2}+y^{2} < 4\) is a circle centered at the origin with a radius of 2. It is a 'less than' inequality, so the solution includes all points inside the circle but not on the circle itself. This area should be shaded to indicate it is included in the solution set.

Step 2: Graph the second inequality

The second inequality \(y - x^{2} \geq 0\) or \(y \geq x^{2}\) represents the region including the parabola \(y = x^{2}\) and all points above it. The 'greater than or equal to' inequality does include the parabola itself. This area should also be shaded.

Step 3: Identify the solution set

The solution set of the system of inequalities includes the points that satisfy both inequalities. This would be the region where the shading of the two graphs overlap. This overlapped region represents the points that satisfy both inequalities.

Key Concepts

Inequality Graphing

Graphing inequalities is a fundamental concept in understanding solutions to systems involving two variables. To graph an inequality, we first treat it as an equation to find its boundary, which can be a line, a curve, or another geometric shape. In the given exercise, we encounter two types of boundaries: a circle for the first inequality and a parabola for the second.

When graphing inequalities, it's important to use a dashed line or curve to indicate that the boundary is not part of the solution when the inequality is strict (e.g., < or >), and a solid line or curve when the boundary is included (e.g., ≤ or ≥). After plotting the boundaries, the next step is to determine the solution region by shading the appropriate area. For the inequality \( x^2+y^2<4 \), we shade the interior of the circle, while for \( y \text{–} x^2 \text{≥} 0 \), we shade the area above and including the parabola. The intersection of these shaded regions gives us the solution set for the system of inequalities.

Quadratic Inequalities

Quadratic inequalities, such as \( y \text{–} x^2 \text{≥} 0 \), include a quadratic expression and can take various shapes on a graph, like parabolas. Understanding their graph involves recognizing if the parabola opens upward or downward and whether the region of interest lies above or below the curve.

To graph a quadratic inequality, we first find the vertex and the axis of symmetry. For \( y = x^2 \), the vertex is at the origin, and the axis of symmetry is the y-axis. When an inequality is 'greater than or equal to', such as our second inequality, it represents the area above or on the parabola. It's essential to identify the correct region since this affects the overlap with other inequalities in a system and influences the solution set.

Circle Equations

Circle equations in the form \( x^2 + y^2 = r^2 \) depict circles on the coordinate plane with a radius \( r \) and centered at the origin. The equation \( x^2+y^2<4 \) represents a circle centered at (0,0) with radius 2. Unlike the equality case where the boundary circle is included (denoted with a solid line), this strict inequality excludes the boundary, hence the need to use a dashed line for the circle.

Students should remember that the inequality \( < \) corresponds to the interior of the circle – a vital concept for identifying the correct region to shade. Graphing circles is often involved in exercises with systems of inequalities as it nicely demonstrates the geometric aspect of solutions involving areas and boundaries.

Algebraic Solutions

Algebraic solutions involve manipulations and calculations to solve equations and inequalities. In the context of system of inequalities, the algebraic approach helps determine the equations of boundaries for graphing. It is also used when analyzing algebraic expression to understand how one variable depends on the other.

For example, re-writing the second inequality as \( y \text{≥} x^2 \) clarifies that for every x-value, the y-value is at or above the parabola. Expressing inequalities algebraically not only guides the graphing process but also aids students in reasoning about the nature of the solutions and the overlapping regions that satisfy all conditions of the system.