91Ó°ÊÓ

The slope-intercept form is a special way of writing the equation of a line so that you can see the slope and y-intercept at a glance. This form is presented as \(y = mx + b\), where \(m\) represents the slope and \(b\) is the y-intercept.
In our original exercise, this form helps to graph the inequality \(4x + 3y \leq 12\). By rewriting it in slope-intercept form, we derived \(y \leq -\frac{4}{3}x + 4\). This transformation helps to clearly identify the slope as \(-\frac{4}{3}\) and the y-intercept as 4.

• Slope \((-\frac{4}{3})\): Represents the steepness and direction of the line. For every 3 units moved horizontally to the right, the line falls by 4 units vertically.
• Y-intercept (4): The point where the line crosses the y-axis.

Transforming inequalities into this form is essential for graphing, as it lays the foundation for outlining the boundary line.

Once the inequality is in slope-intercept form, the next step is to graph the boundary line. This line is crucial because it separates the "solution" area from the "non-solution" area on the graph.
You start by plotting the y-intercept. In this exercise, the y-intercept is 4, meaning the line crosses the y-axis at (0, 4). By using the slope \(-\frac{4}{3}\), you'll plot the next point: move down 4 units and 3 units to the right from (0, 4), resulting in another point at (3, 0).
Having multiple points allows you to draw a precise line. Connect these plotted points directly with a solid line, as the inequality symbol \(\leq\) signifies that the points on the line are included in the solution.
This line serves as a boundary, marking the edge of the region where the solutions to the inequality lie.

The solution area is where our inequality holds true, and we show it through shading on the graph. For our exercise, the inequality \(y \leq -\frac{4}{3}x + 4\) means that all points below the line are solutions.
After drawing the boundary line, the next task is deciding which side of this line to shade. To determine this, choose a test point not on the line—commonly (0, 0) if it's not on the line—and see if it makes the inequality true. Here, substituting (0, 0) into the inequality gives:
\(0 \leq -\frac{4}{3}(0) + 4\) holds true as 0 is less than 4, affirming the region containing this point is the solution area.
This shaded area visually represents the part of the graph where all points satisfy the inequality, making it easy to understand which regions are solutions.