91Ó°ÊÓ

When graphing linear inequalities, the slope-intercept form is a student's best friend. It not only simplifies the process of graphing but also makes it intuitive to understand the direction and the position of the line on the graph. The slope-intercept form is represented by the equation
\( y = mx + b \)
where
\( m \) is the slope and \( b \) is the y-intercept. The slope tells us how steep the line is and in which direction it goes — if it's positive, the line goes upwards when moving from left to right; if negative, it goes downwards. The y-intercept gives you the exact point where the line crosses the y-axis.

Looking at our exercise, rearranging the inequality into slope-intercept form gives us the clear picture of the slope (\(\frac{1}{3}\)) and its y-intercept (\(\frac{5}{3}\)). With these values, you can easily plot the initial point on the y-axis and use the slope to find another point, then draw your line through these points.

The concept of a boundary line is crucial when dealing with linear inequalities. The boundary line is the division between the area where the inequality is true and where it is not. It corresponds to the equality part of the inequality (as if the \( \<\), \( \<=\), \( \>\), or \( \>\=\) sign were just an equals sign).

In our exercise, we show this boundary by drawing the line for the equation \( y = \frac{1}{3}x + \frac{5}{3} \). Here's an important nuance: if the inequality includes the equal sign (as in \( \<=\)) or (\( \>\=\)), the boundary line is solid, showing that points on the line are part of the solution set. If the inequality does not include the equal sign, the line would be dashed, indicating the points on the line are not solutions to the inequality. Consequently, our boundary line in the exercise should be solid, indicating that points lying on the line satisfy the inequality.

The solution area of an inequality represents the set of all points that satisfy the inequality. Once the boundary line is drawn, we need to determine which side of the line contains the solutions. This is done by selecting a test point not on the boundary line. Generally, the origin (0,0) is the simplest test point to use, unless it lies on the boundary line itself.

In our exercise, after testing the origin with the inequality \( y \<= \frac{1}{3}x + \frac{5}{3} \), we find out that the inequality holds true. Therefore, the region including (0,0) is the solution area and we shade this region to indicate all the points that satisfy the inequality. It’s a visual representation of all the possible solutions where any point in this shaded area makes the inequality true.