Problem 12 - A Graphical Approach to Precalculus with Limits

Question

Graph each inequality. $$x-y \geq 2$$

Step-by-Step Solution

Answer

Graph the line \(x - y = 2\), then shade the region above it.

Step 1: Rewrite the Inequality

Start by writing the inequality as an equation for the line: \(x - y = 2\). This will help us find the boundary line that separates the region of solutions.

Step 2: Find the Boundary Line

Determine the intercepts of the line \(x - y = 2\) to graph it. When \(x = 0\), \(y = -2\); when \(y = 0\), \(x = 2\). These intercepts are \((0, -2)\) and \((2, 0)\).

Step 3: Graph the Boundary Line

Plot the points (0, -2) and (2, 0) on a graph and draw a solid line through these points. The solid line indicates that the points on the line are included in the solution (because the inequality symbol is \(\geq\)).

Step 4: Test a Point

Choose a test point that is not on the line to determine which region contains the solutions. A common choice is the origin, \((0, 0)\). Substitute into the inequality: \(0 - 0 \geq 2\). This simplifies to \(0 \geq 2\), which is false.

Step 5: Shade the Correct Region

Since \((0, 0)\) does not satisfy the inequality \(x-y\geq 2\), shade the region *above* the line (or in the direction away from (0,0)) where the inequality holds true.

Key Concepts

Boundary Line

When graphing inequalities, the first major step is to find the boundary line. This line is derived from rewriting the inequality as an equation. In our example, we rewrite the inequality \(x - y \geq 2\) as \(x - y = 2\). The boundary line is crucial because it visually separates the coordinate plane into different regions. To graph this line, identify key points where it crosses the axes, known as intercepts:

• When \(x = 0\), solving \(0 - y = 2\) gives \(y = -2\).
• When \(y = 0\), solving \(x - 0 = 2\) gives \(x = 2\).

These intercepts, \((0, -2)\) and \((2, 0)\), are essential for plotting the boundary line accurately on the coordinate plane. Since the inequality is \(\geq\), the line will be solid, indicating that points on the line satisfy the inequality along with those in the solution region.

Test Point Method

After finding and plotting the boundary line, it's time to determine which side of this line contains the solutions to the inequality. The test point method is a simple and effective way to achieve this.To use this method, pick a point that is not on the boundary line. A popular choice is the origin \((0, 0)\), as long as it isn’t on the line. Substitute this point into the original inequality:

• For the inequality \(x - y \geq 2\), substitute \((0, 0)\):
• \(0 - 0 \geq 2 \rightarrow 0 \geq 2\)
• The statement is false.

Since the inequality does not hold true for \((0,0)\), this means the solution lies in the region on the opposite side of the plane from the test point.

Coordinate Plane

The coordinate plane is the playing field where we graph the inequality. It consists of two axes: a horizontal \(x\)-axis and a vertical \(y\)-axis, intersecting at the origin \((0, 0)\).When plotting an inequality, understanding the coordinate system helps in accurately identifying points and regions. The plane is divided by the boundary line into two areas, each representing different sets of solutions. One region will satisfy the inequality, while the other won't.With our example line \(x - y = 2\), the points (0, -2) and (2, 0) are plotted on this plane. Correct interpretation and plotting on the coordinate plane are key to ensuring the shaded solution region accurately represents the inequality.

Solution Region Shading

Shading the solution region is the last step in graphing an inequality. Once you've tested a point to determine which side of the boundary line contains valid solutions, you shade that region.In our exercise:

• Since the point \((0, 0)\) did not satisfy \(x - y \geq 2\), the solution region is the area above the line \(x - y = 2\).
• Shading indicates that any point in this region will satisfy the inequality.

The type of line used is vital; since the inequality symbol is \(\geq\), a solid line is drawn to show inclusion of boundary points. Shading provides a visual cue, ensuring anyone looking at the graph can easily identify all points that satisfy the inequality.