91Ó°ÊÓ

The slope-intercept form is a very handy tool in graphing linear equations or inequalities. It's written as \(y = mx + b\). Here, \(m\) represents the slope of the line—showing how steep the line is—and \(b\) indicates where the line intersects the y-axis, known as the y-intercept.

The slope, \(m\), tells us just how much the line moves up or down as we move along the x-axis. If it's positive, the line tilts upward, and if it's negative, the line tilts downward. The y-intercept, \(b\), is the starting point on the y-axis.

For our exercise, the equation \(y = -\frac{3}{4}x - \frac{1}{2}\) simplifies everything. Here, \(m = -\frac{3}{4}\) means the line goes down 3 units for every 4 units it moves right, and \(b = -\frac{1}{2}\) tells us it crosses the y-axis below the origin.

In the context of inequalities, the boundary line acts as a sort of fence for the graphical solution. It tells us the dividing line between regions that do and do not satisfy the inequality.

In our exercise, we derived the equation \(3x + 4y = -2\) from the inequality \(3x + 4y \leq -2\). By graphing this boundary line, we sketch out the line that will help us decide where the solutions lie.

To draw it, start at the y-intercept which is \(-\frac{1}{2}\), then use the slope of \(-\frac{3}{4}\) to find another point. Draw a solid line through these points, indicating that points on the line are part of the solution set (because of the 'equal to' part of \(\leq\)).

After sketching the boundary line, the next step is determining which side of the line satisfies the inequality. This process involves shading the region.

In this example, because we used a \(\leq\) sign, we identify the area that includes and goes beyond the boundary line. Using a test point like the origin helps us. If the point doesn't satisfy the inequality, shade the opposite side.

Since our test point (0,0) suggested that the inequality doesn't hold true (as \(0 \leq -2\) is false), it means the area directly across the line holds the solutions. Therefore, you would shade above the line. This shaded region shows all the sets of \((x, y)\) pairs that satisfy the original inequality.

Test points are a simple yet essential method for confirming which region satisfies an inequality on a graph. The most common choice is the origin point (0,0), unless it lies on the boundary line itself.

Plugging the coordinates into the inequality helps to figure out where the solutions lie. If the inequality holds true, the region containing the test point is shaded. If false, the region opposite to the test point is the solution region.

In our example, substituting (0,0) into \(3x + 4y \le -2\) results in an incorrect statement (0 is not less than or equal to -2). Hence, the correct region to shade is not the one containing the origin, but its opposite side relative to the boundary line.