Problem 27 - Introductory Algebra for College Students

Question

Graph the solution set of each system of linear inequalities. $$\left\\{\begin{array}{l}x \geq 0 \\\y>0\end{array}\right.$$

Step-by-Step Solution

Answer

The solution to these system of inequalities is the region which lies in the first quadrant, excluding the x-axis.

Step 1: Understand the individual Inequalities

Our inequalities \(x \geq 0\) and \(y > 0\) tells us that the values which satisfy these inequalities lies in the quadrant where x is positive or equals to zero and y is strictly positive. The inequality \(x \geq 0\) will be a vertical line passing through the origin, while \(y > 0\) will be a horizontal line also passing through the origin.

Step 2: Graph the first inequality \(x \geq 0\)

Graph a vertical line passing through the origin (0,0). Because the inequality includes equals sign, the line itself is included in the solution and therefore we make a solid line. The region which satisfy this inequality will be to the right side of the line (includes the line itself).

Step 3: Graph the second inequality \(y > 0\)

Graph a horizontal line passing through the origin (0,0). Because the inequality \(y > 0\) does not include equals sign, the line itself is not included in the solution. Hence, we make a dashed line. The region which satisfy this inequality will be the upper side of this line not inclusive of line itself.

Step 4: Find the solution set

The solution to the system of inequalities will be the area of overlap from all inequalities graphed. In this case, the solution would be everything above the x-axis (exclusive) and everything to the right of y-axis (inclusive). Therefore, we only shade the region which lies in the first quadrant, excluding the x-axis.

Key Concepts

System of Inequalities

A system of inequalities consists of two or more inequalities that you consider simultaneously. In order to find a solution, we look for all possible pairs of values that satisfy every inequality in the system.
When working with a system of inequalities involving only linear equations, the solutions might form specific regions in a graph. This is unlike a single inequality which simply divides the coordinate plane into a shaded region and an unshaded region.
For example, consider our system of inequalities:

• For the inequality \(x \geq 0\), it implies that x can be zero or any positive number.
• For the inequality \(y > 0\), it includes all positive values of y.

Thus, the solution to the system includes all the points (x, y) that satisfy both conditions. In essence, this means covering the section of the graph where these x and y values occur simultaneously. Understanding these intersections provides insight into what parts of the coordinate plane hold the possible solutions for given inequalities.

Quadrants in Coordinate Plane

In a coordinate plane, the space is divided into four sections by the x-axis and y-axis, and these sections are called quadrants. Each quadrant houses different types of coordinate pairs.
Here’s how they are characterized:

• **First Quadrant**: Both x and y are positive.
• **Second Quadrant**: x is negative while y is positive.
• **Third Quadrant**: Both x and y are negative.
• **Fourth Quadrant**: x is positive while y is negative.

In our exercise, the valid region for the system of inequalities \(x \geq 0\) and \(y > 0\) is the first quadrant. This is because it's the only quadrant where both x is zero or positive, and y is strictly positive. The x-axis behaves like a boundary, only being touched at the y-axis, while the y-axis itself can be part of the solution.

Graphical Solution Methods

A major part of solving systems of linear inequalities involves graphing and finding the shaded regions. This is often termed graphical solution methods: interpreting the graphic representation of inequalities.
Here's how you can approach it effectively:

• The first step is graphing each inequality line on the coordinate plane. For example, \(x \geq 0\) is drawn as a vertical line at x = 0 and \(y > 0\) is a horizontal line at y = 0.
• Ensure clarity in the graph by knowing whether to draw a 'solid' or 'dashed' line: a solid line when the line is part of the solution (\(\geq\) or ≤),and a dashed line when it’s not (\(>\) or <).
• The overlapping area of the shaded regions for all inequalities is your solution set.

In our case, you'll find that the solution set is above the x-axis and right of the y-axis but doesn't include the x-axis line itself since it is dashed for the inequality \(y > 0\). Understanding these methods helps visualize regions of interest for solutions, connecting algebraic conceptswith their graphical interpretations.