Problem 37 - Introductory Algebra for College Students

Question

Graph the solution set of each system of linear inequalities. If the system has no solutions, state this and explain why. $$\left\\{\begin{array}{l}x+y \geq 1 \\\x-y \geq 1 \\\x \geq 4\end{array}\right.$$

Step-by-Step Solution

Answer

The solution of this system of inequalities is the region right to the line \(x = 4\) and above both lines \(x + y = 1\) and \(x - y = 1\).

Step 1: Graph each inequality separately

Start by sketching the graph for each inequality separately. First, treat it as a regular linear equation by ignoring the inequality sign:For \(x + y = 1\), it is a linear equation with a slope of -1 and y-intercept of 1. Draw the line on the Cartesian plane.Similarly for \(x - y = 1\), the slope is 1 and y-intercept is -1. Draw this line too.Finally \(x = 4\) represents a vertical line crossing the x-axis at 4.

Step 2: Determine the solution region for each inequality

Once the lines are drawn, shade the region which satisfies the inequality for each line:For \(x + y \geq 1\), shade the region above the line, including the line because of \(\geq\).For \(x - y \geq 1\), shade the region above the line, including the line.For \(x \geq 4\), shade the region right to the line, including the line because of \(\geq\).

Step 3: Find the common solution region

The solution of the system of inequalities is the intersection of all the regions. This can be found by looking at where all the shaded areas overlap. In this particular case, it is the region right to the line \(x = 4\) and above both lines \(x + y = 1\) and \(x - y = 1\).\

Step 4: No Solution Case

If there was no region where all conditions were satisfied, it would mean the system has no solution. This would occur if the shaded regions for each line did not overlap. However, in this case, a solution exists, identified in step 3.

Key Concepts

Graphing Linear Inequalities

Graphing linear inequalities involves plotting the lines which represent the equations of the inequalities, then shading one side of these lines to show where the inequality holds true. It commences with viewing each inequality as if it were a standard linear equation by simply ignoring the inequality sign for the moment. For example, for the inequality \(x + y \geq 1\), we initially treat it as \(x + y = 1\) which is a line with a slope of -1 and a y-intercept of 1.
Once you have plotted the corresponding line, you then determine which side of the line the inequality represents by testing a point that is not on the line (often the origin is used unless it lies on the line).

• If the point satisfies the original inequality, you shade the side of the line containing that point.
• Otherwise, shade the opposite side.

It's vital to remember that if the inequality is \(\geq\) or \(\leq\), the line itself is part of the solution, so it should be drawn solid. However, for \(>\) or \(<\), the line is not part of the solution and should be dashed.

Intersection of Regions

The intersection of regions is a crucial concept when solving systems of linear inequalities. Each inequality you graph will produce a shaded region that represents all the possible solutions for that particular inequality.
Imagine that each of these regions is a different color drawn on a transparent piece of paper. The solution to the system of inequalities is where all the colors overlap.
This overlapping area is where all the conditions from each inequality in the system are satisfied simultaneously.

• If an area on the graph satisfies each inequality's condition, it is the intersection of their regions and hence part of the solution set.
• If no such overlapping area exists, then the system of inequalities has no solution.

In our example, the common region that satisfies all three inequalities \(x + y \geq 1\), \(x - y \geq 1\), and \(x \geq 4\) is identified by the overlap of all shaded regions. In cases where there is no overlapping region, it indicates there is no simultaneous solution to the inequalities.

Understanding Linear Equations

Linear equations are the foundation for graphing inequalities. They represent a straight line on a coordinate plane and are often expressed in the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
Understanding them is key because they form the boundary lines for linear inequalities. The slope \(m\) tells us how steep the line is, with positive slopes ascending from left to right and negative slopes descending.
The y-intercept \(b\) shows where the line crosses the y-axis, which is a critical starting point for graphing. In the process of transforming an inequality into a linear equation, you focus on the equality part, allowing you to simply sketch its boundary line.

• The slope and intercept make it easy to quickly draw the line on a graph.
• The equation gives a clear rule for any point that lies on the line.

Once you have fully grasped the concept of linear equations, the next step is to apply it to graphing inequalities, which involves shading the relevant side of these boundary lines to represent inequality solutions.