Problem 41 - College Algebra

Question

Graph the solution set of each system of inequalities or indicate that the system has no solution. $$ \begin{aligned}&y \geq x^{2}-1\\\&x-y \geq-1\end{aligned} $$

Step-by-Step Solution

Answer

The solution set is the region that falls both on the parabola above \(y = x^{2}-1\) and on/above the line \(x-y =-1\). This region is marked by shading on the graph.

Step 1: Plot Inequality 1

The inequality \(y \geq x^{2}-1\) is a parabola that opens upwards with a vertex at (0,-1). Plot this parabola. Points on the parabola and above it will satisfy this inequality.

Step 2: Plot Inequality 2

The inequality \(x-y \geq -1\) is a line with slope 1 and intersects the y-axis at 1. Plot this line. Points that are above and to the right of this line will satisfy this inequality.

Step 3: Find Common Region

Look for the region where both inequalities are satisfied. This will be the area that falls both on the parabola above \(y = x^{2}-1\) and on/above the line \(x-y =-1\). This common region represents the solution set for our system of inequalities.

Step 4: Mark the Solution Set

Shade the common region to indicate the solution set. Any point in this shaded region is a solution to the system of inequalities.

Key Concepts

Graphing Inequalities

Graphing inequalities involves visually representing the solutions to inequalities on a coordinate plane. When you plot the graph of an inequality, you are showing all the possible values that satisfy the inequality.
First, identify if the inequality involves a parabola or a line by examining the expression. A quadratic expression, like \( y \geq x^2 - 1 \), indicates a parabola. For linear inequalities, such as \( x - y \geq -1 \), the expression is a line.
When graphing a parabola, such as \( y \geq x^2 - 1 \), you should remember:

• Plot the parabola by finding its vertex and axis of symmetry.
• Shade the region above the parabola, as the inequality \( \geq \) means greater than or equal to.

For a linear inequality like \( x - y \geq -1 \):

• Find the y-intercept and slope to plot the line. In this case, the slope is 1, and the line intersects the y-axis at \( y = 1 \).
• Shade the area above and to the right of the line, because points in this region satisfy the inequality.

Graphing in this way helps you visually determine the set of all solutions at a glance.

Solution Set

The solution set of a system of inequalities is the collection of all points that satisfy every inequality in the system. To find the solution set, you must identify the region where the solutions to the individual inequalities overlap.
In practice:

• First, graph each inequality on the same coordinate plane.
• Look for the overlapped area where the shaded regions of each individual inequality meet. This region is your solution set.
• Check whether the border lines of the inequalities should be included in the solution set by looking at the inequality symbols used. Here, because both inequalities use \( \geq \), the border lines are part of the solution set.

The solution set represents all the possible values that solve the system, practically showcasing how single variables add up to meet the system of inequalities.

Parabola and Line Intersection

Understanding the intersection between a parabola and a line is crucial when solving a system of inequalities that involve both. The intersection points are where the line crosses the parabola, creating the boundaries for the regions being shaded.
For our example, the inequality \( y \geq x^2 - 1 \) describes a parabola that opens upwards with a vertex at \( (0, -1) \).
The inequality \( x - y \geq -1 \) describes a line. When these two graphs intersect, the solution set will be the area above both the parabola and the line.
To find the point of intersection:

• Set the equations of the line and parabola equal: \( x^2 - 1 = x + 1 \).
• Solve for \( x \) to find where these graphs meet. This helps define the boundary of your solution set.
• Finally, plot these points on the graph and shade the region covered by the solutions of both inequalities.

This intersection visually determines where both conditions are satisfied, leading you to the final solution set. Identifying these points helps provide the complete picture of what's happening in the system.