Problem 46 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

The solution to the system of inequalities is the region where the shaded areas of both inequalities intersect.

Step 1: Graph the first inequality

Firstly, analyze the inequality \(y \geq x^{2}-4\). This represents a parabola opening upwards with a vertex at (0, -4). Graph this parabola and then, because of the 'greater than or equal to' sign, shade the region above the parabola.

Step 2: Graph the second inequality

Now, graph the inequality \(x-y \geq 2\). Rewrite this inequality in slope-intercept form as \(y \leq x - 2\) which represents a line with a y-intercept at -2 and slope of 1. After graphing the line, shade the region below it due to the 'lesser than or equal to' sign.

Step 3: Find the intersection of the two shaded regions

The solution to the system of inequalities is the overlapping region of the two shaded areas on the graph plotted in the previous steps. This region represents all points (x, y) satisfying both inequalities.

Key Concepts

Graphing Inequalities

Graphing inequalities involves visualizing areas on a coordinate plane that satisfy a given inequality. To start graphing an inequality, you need to consider the equality part first. For example, with the inequality \(y \geq x^{2} - 4\), the equality \(y = x^{2} - 4\) is graphed as a regular parabola. Once the basic curve or line is drawn, shading is essential. Here, because of the 'greater than or equal to' sign, you shade above the parabola, covering all greater values of \(y\).
Imagine the shaded area as all the possible solutions that make the inequality true. For systems of inequalities, as in the exercise, multiple shaded regions can overlap. This overlapping region contains all the solutions that satisfy all inequalities in the system, which is crucial to finding the solution set. Using dotted or solid lines is important too. A solid line shows that points on the line are included (\( \geq \) or \( \leq \)), whereas a dotted line indicates they are not included (\( > \) or \( < \)).
Graphing inequalities improves problem visualization, making it easier to understand and solve complex systems.

Parabolas

Parabolas are symmetric curves represented by quadratic equations like \(y = x^2 + bx + c\). The example \(y = x^2 - 4\) is a simple parabola opening upwards with its vertex at (0, -4). A parabola's direction of opening depends on the sign of the coefficient in front of \(x^2\). If positive, the parabola opens upwards, and if negative, it opens downwards.
The vertex is a significant point. It is either the lowest point if the parabola opens upwards, or the highest point if it opens downwards. The axis of symmetry, a vertical line through the vertex, divides the parabola into two mirror-image halves. You can find the axis by setting \(x = -\frac{b}{2a}\), but in simpler forms like \(y = x^2 - 4\), it is simply the y-axis (\(x = 0\)).
To graph a parabola accurately, identify additional points on either side of the vertex. This can help in drawing a more precise curve, which is important when combining with other graph lines in systems of inequalities.

Linear Equations

Linear equations, like \(x - y = 2\), form straight lines when graphed. Each linear equation can be expressed in slope-intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. The slope indicates the steepness and direction of the line, while the y-intercept is the point where the line crosses the y-axis.
In this exercise, the inequality \(x - y \geq 2\) converts to \(y \leq x - 2\). Therefore, the line starts at \(b = -2\) with a slope of 1, meaning it rises one unit for every one unit it moves to the right. The line separates the plane into two regions, and since the inequality is 'less than or equal', the solution region is beneath the line. Like curvy parabolas, the choice of shading highlights the area of valid solutions.
Understanding and graphing linear equations is fundamental in solving systems of inequalities, as the points of intersection with other lines or curves reveal solutions that satisfy multiple conditions simultaneously.