Problem 14 - Elementary Algebra Within Reach

Question

In Exercises \(7-16\), sketch the graph of the system of linear inequalities. $$ \left\\{\begin{array}{l} y \leq-x \\ y \leq x+1 \end{array}\right. $$

Step-by-Step Solution

Answer

The graph of this system of linear inequalities consists of two solid lines, one sloping downwards from the origin with shading underneath it, and the other sloping upwards from the point (0,1) also with shading underneath it. The region where these shadings overlap indicates the solution set for the system of inequalities.

Step 1: Graphing the first inequality

First, identify the inequality \(y \leq -x\). This represents a line with a slope of -1 passing through the origin (0,0). The line should be solid, as it includes points where \(y = -x\). Also, since \(y\) is less than or equal to \(-x\), shade the area beneath the line.

Step 2: Graphing the second inequality

Next, identify the inequality \(y \leq x+1\). This line has a positive slope of 1 and a y-intercept at (0,1). The line should also be solid, as it includes points where \(y = x+1\). Again, because \(y\) is less than or equal to \(x+1\), shade the area beneath the line.

Step 3: Finding the overlap area

The system of inequalities is satisfied where the shading overlaps. This represents the set of points that satisfy both inequalities simultaneously. Typically the overlapping area will form a polygon, but in this case, as the slopes of both the functions are equal, the overlapping part will look like a triangular sort of region.

Key Concepts

System of Inequalities

A system of inequalities consists of two or more inequalities that are considered simultaneously. To solve such a system means finding all the sets of values that satisfy every inequality in the system. When graphing these in a coordinate plane, each inequality divides the plane into two regions. Typically, one region is a solution set and can be determined by shading. By considering the system as a whole, the solution is the area where all these shaded regions overlap.
This can be visualized as creating constraints. Each inequality sets a limit, forming boundaries on a graph. In our given exercise, the inequalities are:

• \(y \leq -x\)
• \(y \leq x + 1\)

The combined solution set will be where these individual solution sets intersect.

Slope

The slope, often represented by "m" in the equation of a line \(y = mx + b\), describes the steepness and direction of the line. It is calculated as the ratio of the change in the y-coordinate to the change in the x-coordinate. This is often phrased as "rise over run."

• For the inequality \(y \leq -x\), the slope of the line is -1. This means for every unit increase in x, y decreases by 1; hence, the line moves downward as it goes from left to right.
• For the inequality \(y \leq x + 1\), the slope of the line is 1. This indicates that for every unit increase in x, y also increases by 1, making the line ascend as it moves from left to right.

Understanding slopes is essential, as it not only helps in drawing the line accurately but also conveys the rate and direction of change of the relationship depicted.

Shading Regions

Shading helps visualize the regions that satisfy the inequality. Once the line corresponding to an inequality is drawn, the region that represents the solution set for the inequality is shaded.

• For the inequality \(y \leq -x\), the solution is every point on or below the line. This is why the area beneath this line is shaded.
• Similarly, for \(y \leq x + 1\), the area below this line needs to be shaded. This region represents all points where \(y\) is less than or equal to \(x + 1\).

The shading helps highlight the part of the graph where both inequalities are true. In this exercise, the shaded regions of both inequalities will overlap, showing us where the system of inequalities holds true.

Graph Intersection

The graph intersection is where the shaded regions of different inequalities overlap. In the context of graphing inequalities, this region represents the common solution to the system of inequalities.
When graphing the system:

• The line for \(y \leq -x\) will intersect with the line for \(y \leq x + 1\) somewhere in the coordinate plane.
• The overlapping shaded areas indicate the set of points that satisfy both inequalities at once.
• In our exercise, the intersection forms a triangular region below and between these lines.

This intersection is crucial because it visually represents the solution set for the system, helping us quickly identify all possible solutions that satisfy both conditions imposed by the inequalities.