Problem 42 - Precalculus with Limits

Question

In Exercises 33-46, sketch the graph (and label the vertices) of the solution set of the system of inequalities. $$\left\\{\begin{array}{l}{x-y^{2}>0} \\ {x-y>2}\end{array}\right.$$

Step-by-Step Solution

Answer

The solution set for the system of inequalities lies in the region right to the parabola of equation \(x=y^2\) and the straight line \(x=y+2\), where these two areas intersect.

Step 1: Convert Inequalities to Equations

Convert the inequality \(x-y^2>0\) to the equation \(x=y^2\). Similarly, convert \(x-y>2\) to \(x=y+2\). In the equation \(x=y^2\), y can be negative or positive, and hence it represents a parabola. While the equation \(x=y+2\) represents a straight line.

Step 2: Graph the Equations

Plot the resulting equations on the coordinate plane. For \(x=y^2\), plot values for y ranging from -3 to 3 and find corresponding x values. This will give a parabola opening towards positive x-axis. For the equation \(x=y+2\), for y=-1, x=1 so the line crosses the x-axis at (1,0); similarly for x=-1, y=-3 so the line crosses the y-axis at (0,-3).

Step 3: Finding Solution Areas

The first inequality, \(x-y^2>0\), means that we are looking for values of x that are greater than \(y^2\), hence the solution lies to the right of the plotted parabola. The second inequality, \(x-y>2\), means the solution lies right to the line \(x=y+2\). The intersection of these two regions will give the shared solution set.

Key Concepts

Graphing Inequalities

To solve systems of inequalities graphically, start by treating each inequality as an equation. For example, to solve the system \(x - y^2 > 0\) and \(x - y > 2\), first rewrite them as equations. This allows you to graph them as a parabola and a straight line respectively. Once the equations are graphed, identify areas fulfilling the inequality conditions. Remember, inequalities like \(x-y^2 > 0\) imply that we seek values where \(x\) is greater than \(y^2\). Similarly, \(x-y > 2\) implies \(x\) should be more than \(y + 2\). By focusing on these regions, you can identify the correct sections of the graph that meet all inequality conditions. - Use dotted lines if the inequality is strict (\(>\) or \(<\)).- If inequalities were non-strict (\(\ge\) or \(\le\)), use solid lines.- Shade overlapping regions that satisfy all inequalities to find the solution set.

Parabola

In our system, the inequality \(x - y^2 > 0\) translates to the equation \(x = y^2\), forming a parabola. A parabola is a curve that is symmetric about a central axis. Here, this parabola opens horizontally to the right because \(x\) is dependent on \(y^2\), and \(x\) values are greater than zero. - To sketch this parabola, choose a range of \(y\) values (e.g., \(-3, -2, -1, 0, 1, 2, 3\)).- Calculate the corresponding \(x\) values.- Plot these coordinates: \((y^2, y)\) on the graph.- Remember, the area satisfying \(x-y^2 > 0\) will be to the right of the parabola.

Linear Equations

Linear equations create straight lines when graphed. The inequality \(x - y > 2\) rewrites to the equation \(x = y + 2\), forming a straight line. Straight lines are characterized by having a constant slope, which makes them simple to model on a coordinate plane.- Find points using the slope-intercept form, \(y = mx + c\), which in this case transforms to \(x = y + 2\).- Gather a few \(y\) values to find corresponding \(x\) values, like \(y = -1\), \(x = 1\) giving the point \((1,-1)\).- Plot enough points to accurately draw the line.- Since \(x - y > 2\), shade the area to the right of this line to represent \(x\) values greater than \(y + 2\).

Solution Set

The solution set of a system of inequalities is the region where the solutions of each inequality overlap. Once you have graphed your parabola and line, focus on finding the intersection area that satisfies both conditions simultaneously:- For \(x - y^2 > 0\), shade the area to the right of the parabola.- For \(x - y > 2\), shade the area to the right of the line.- The solution set is the intersection of these shaded areas, forming a distinct region on the graph.This resulting intersection is the solution set, where both inequalities are true. It illustrates all possible solutions to the system, enabling better understanding and visualization of these mathematical concepts.