91Ó°ÊÓ

Before graphing inequalities, it's important to rewrite them in slope-intercept form. This format is given by:

y = mx + b, where:

• y is the output (dependent variable)
• m is the slope (rate of change)
• x is the input (independent variable)
• b is the y-intercept (where the line crosses the y-axis)

Slope-intercept form makes it easier to graph since you can quickly identify the slope and y-intercept.
In the provided exercise, the inequalities:

• $$ 3x + 2y < 6 $$
• $$ x - 2y > 2 $$

are turned into slope-intercept format:

• $$ y < -\frac{3}{2}x + 3 $$
• $$ y < \frac{1}{2}x - 1 $$

This arrangement enables easy graphing on a coordinate plane as you'll see in the next sections.

After placing the inequalities into slope-intercept form, the next step is to graph them. A key part of graphing inequalities is shading the correct regions.

For the inequality:

• $$ y < -\frac{3}{2}x + 3 $$
• $$ y < \frac{1}{2}x - 1 $$

You draw the lines as boundaries:

• -\frac{3}{2}x + 3 and \frac{1}{2}x - 1

Typically, you'll draw a dashed line (more on that later).
Then, you shade below these lines because the inequalities use '<'.

Shading visually represents all the points (x, y) that satisfy the inequality. It's like marking the area where the solution lives. When dealing with two or more inequalities, pay attention to whether you're asked for the union ('or') or the intersection ('and') of the regions. Here, since we want the union, any point in either shaded region counts as a solution.

When graphing inequalities, the type of line you draw—dashed or solid—plays an important role.

In the given exercise, both inequalities use '<' signs:

• $$ y < -\frac{3}{2}x + 3 $$
• $$ y < \frac{1}{2}x - 1 $$

A dashed line represents that the boundary line itself is not included in the solution.
This is because '<' means 'less than,' not 'less than or equal to'. So every point exactly on the line is not part of the solution.
Additionally, remember that inequalities with '>' or '<' always use dashed lines, while '≥' or '≤' use solid lines.
solid lines include the boundary as part of the solution.

Using a dashed line helps to visually differentiate between what is and isn't included in your solution set. Always pay attention to these small details, as they can make a big difference in your final answer.