Problem 23 - Precalculus with Limits

Question

In Exercises 19-28, use a graphing utility to graph the inequality. $$y \leq 6-\frac{3}{2} x$$

Step-by-Step Solution

Answer

The solution is the graph of the line \(y = 6-\frac{3}{2} x\) and the region below it, including the line itself.

Step 1: Rewrite inequality

Consider the inequality \(y \leq 6-\frac{3}{2} x\). This corresponds to the line \(y = 6-\frac{3}{2} x\). The line is a boundary that separates the coordinate plane into two regions.

Step 2: Graph the line

Using a graphing utility, plot the line \(y = 6-\frac{3}{2} x\). This is a straight line with a slope of \(-\frac{3}{2}\) and a y-intercept of 6. The slope of -3/2 means that for every 2 units you go to the right, you go down by 3 units.

Step 3: Determine which region to shade

To choose which region to shade, take a test point not on the line. The origin (0,0) is usually the simplest. Substitute x=0 and y=0 back into the original inequality \(0 \leq 6 - \frac{3}{2}*0\), which simplifies to \(0 \leq 6\). Because this is true, the region that includes the origin should be shaded.

Step 4: Shading

Shade the region that contains the point (0,0). This means shading below the line.

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is a way to easily identify the slope and y-intercept of a line. It is generally represented as \( y = mx + b \), where:

• \( m \) is the slope of the line.
• \( b \) is the y-intercept, where the line crosses the y-axis.

For the inequality \( y \leq 6 - \frac{3}{2} x \), the equation of the line in slope-intercept form is \( y = -\frac{3}{2}x + 6 \). Here:

• The slope (\( m \)) is \(-\frac{3}{2} \). This tells you that for every 2 units you move to the right along the x-axis, you move 3 units down along the y-axis.
• The y-intercept (\( b \)) is 6. This means the line crosses the y-axis at 6.

Understanding the slope-intercept form helps in sketching a quick graph of the line, which is crucial when graphing linear inequalities.

Shading Regions

Shading regions is a key step when graphing inequalities. It shows the set of all solutions to the inequality. In the inequality \( y \leq 6 - \frac{3}{2} x \), the line \( y = 6 - \frac{3}{2} x \) itself is the boundary.

When shading, follow these simple steps:

• First, plot the boundary line. Since the inequality is \( \leq \), the line is solid. This indicates points on the line are included in the solution set.
• Choose a test point that is not on the line, such as the origin (0, 0). Substitute this point into the inequality.
• If the inequality holds true with the test point, shade the region where the point lies.

In this example, plugging (0, 0) into the inequality \( 0 \leq 6 - 0 \) results in a true statement \( 0 \leq 6 \). Thus, shade the region below the line, including the line itself. Shading helps visualize all the possible solutions to the inequality.

Graphing Utilities

Using graphing utilities such as graphing calculators or software is incredibly useful for visualizing inequalities quickly and precisely. These tools enhance learning by automatically drawing lines and shading areas without manual plotting.

When using graphing utilities:

• Input the inequality directly to see the graph instantly. The utility will display the boundary line and the correct shaded region.
• Watch how adjustments to the inequality change the graph. This interactive feature aids in understanding how different values affect slope and positioning.
• Explore multiple inequalities at once to see their intersections and how they interact.

Graphing utilities save time and minimize errors in complex calculations, allowing more focus on interpreting and understanding the solutions represented by the graph. They are an excellent resource for mastering graphing skills more efficiently.