Problem 16 - Elementary Algebra Within Reach

Question

In Exercises \(7-16\), sketch the graph of the system of linear inequalities. $$ \left\\{\begin{array}{l} x-y>2 \\ x-y<-4 \end{array}\right. $$

Step-by-Step Solution

Answer

The system of inequalities doesn't have a solution because the regions satisfying the individual inequalities do not overlap.

Step 1: Transform Inequalities into Slope-Intercept Form

Rewrite both inequalities into the form y = mx + b. Therefore, we have: 1. \( x - y > 2 \) rewrites to \( y < x - 2 \) 2. \( x - y < -4 \) rewrites to \( y > x + 4 \)

Step 2: Graph First Inequality

Plot the line \( y = x - 2 \) on coordinate axes, which has a slope of 1 and y-intercept at -2. Set this line as a dashed line since the inequality sign '<' points out that the points on the line are not included in the solution set. Lastly, shade the region below this line, signifying that all the points in that region satisfy \( y < x - 2 \).

Step 3: Graph Second Inequality

Sketch the line \( y = x + 4 \), which has a slope of 1 and a y-intercept at 4. As with the previous step, delineate the line as a dashed line because the points on the line are not part of the solution due to inequality symbol '>'. Lastly, shade the area above this line because all the points in that region fulfill \( y > x + 4 \).

Step 4: Identify Solution Region

The solution to the system of inequalities is the overlapping region of the two individual solutions. In this case, there's no overlapping region, indicating that there's no solution to this system of inequalities.

Key Concepts

Graphing Linear Inequalities

Graphing linear inequalities involves more than just sketching lines on a graph. Each inequality divides the coordinate plane into different regions. To represent these inequalities correctly, we usually use dashed or solid lines, depending on the strictness of the inequality.
- **Dashed Lines**: Use dashed lines when the inequality symbols are "<" or ">", indicating that points on the line itself are not included in the solution. - **Solid Lines**: If the inequality is "≤" or "≥", use solid lines to show that points on the line are part of the solution.
After plotting the line, you need to shade the region that satisfies the inequality. An easy way to determine which side to shade is to test a point not on the line (often the origin, if it's not on the line) and see if it makes the inequality true. This helps identify the correct half-plane to shade. When graphing a system of inequalities, the solution region is where the shaded areas overlap.

Slope-Intercept Form

The slope-intercept form is a powerful tool in graphing linear inequalities. It is expressed as: \( y = mx + b \) where:

• m is the slope of the line, indicating its steepness and direction. It tells us how much the y-value changes for a unit change in the x-value.
• b is the y-intercept, representing the point where the line crosses the y-axis.

For inequalities, converting them to slope-intercept form makes graphing easier. You identify the line, plot the y-intercept, apply the slope to find another point, and then draw the line. This form clearly shows how to construct the line graphically and aids in recognizing the nature of the relationship—whether it rises or falls on the coordinate plane.

Solution Region

The solution region in systems of linear inequalities is crucial. It refers to the set of all possible points on the coordinate plane that satisfy all inequalities simultaneously.
Once each inequality is graphed and the appropriate regions are shaded, finding the solution involves identifying the region where these shaded areas overlap. This common area gives the solution set. If there's no overlapping region, the system of inequalities has no solution. In some cases, the solution region might be infinite, extending along a plane, while sometimes it could be a bounded area. Understanding the solution region helps in determining feasible solutions and analyzing systems in real-world problems.

Coordinate Plane

The coordinate plane is the fundamental space where all linear inequalities are graphed. Consisting of two perpendicular number lines—the x-axis and y-axis—it purposes as a grid for plotting points and lines. Each point on this plane corresponds to an ordered pair \((x, y)\), with the x-value representing horizontal positioning, and y-value representing vertical positioning.
To effectively use the coordinate plane, you need to grasp how it is divided into four quadrants, each identified by the signs of the coordinates in that section:

• Quadrant I: \((+,+)\)
• Quadrant II: \((-,+)\)
• Quadrant III: \((-,-)\)
• Quadrant IV: \((+,-)\)

Understanding the layout of the coordinate plane aids in plotting linear inequalities accurately, analyzing intersections, and visualizing their solutions comprehensively.