91Ó°ÊÓ

The slope-intercept form is a way to express linear equations in a format that is easy to graph. This form is given by:\[ y = mx + b \]where:

• \( m \) represents the slope of the line, showing the change in \( y \) for every unit change in \( x \).
• \( b \) is the y-intercept—the point where the line crosses the y-axis, when \( x = 0 \).

Having an inequality like \( 5x + 4y \geq 20 \), the goal is to solve for \( y \) to get it into slope-intercept form. Doing so helps identify both the slope and y-intercept. Here, solving gives us \( y = -\frac{5}{4}x + 5 \).
This makes it easy to visualize and graph the line, using \(-\frac{5}{4}\) as the slope and \(5\) as the y-intercept.

The boundary line in the graph represents the equation obtained when the inequality is turned into an equality. In the case of \( y \geq -\frac{5}{4}x + 5 \), the line \( y = -\frac{5}{4}x + 5 \) serves as the boundary.
For this inequality, the boundary line must be drawn as a solid line, because the inequality sign \( \geq \) includes 'equal to', showing that points on the line are part of the solution set. You start graphing by plotting the y-intercept: the point \((0, 5)\) on the y-axis.
Then use the slope \(-\frac{5}{4}\) to find another point by moving down 5 units and right 4 units. Connect these points with a solid line to complete the boundary line.

Shading regions on a graph visually represents all the solutions of the inequality. For the inequality \( y \geq -\frac{5}{4}x + 5 \), the solutions reside above the boundary line.
Shading above means that all points in that region have a \( y \)-value greater than or equal to what the line equation provides for any \( x \)-value. To ensure you shade the correct area, pick a test point that is not on the boundary line, like the origin \((0, 0)\). Substituting this point into the original inequality, we find:\[ 5(0) + 4(0) \geq 20 \Rightarrow 0 \geq 20 \]This statement is false, confirming our shading should not include this point. As a result, the true shaded region lies above the boundary line, highlighting where \( y \) values are larger for the given \( x \).