91Ó°ÊÓ

Understanding inequalities in two variables is important in algebra. An **inequality** shows the relation between two expressions using inequality symbols like <, >, ≤, or ≥. When we talk about **inequalities in two variables**, it means we're dealing with expressions involving both x and y, such as in the exercise where we had the inequalities:
- \(y \, \geq \, \frac{1}{3} \ x - 2\)
- \(y \, \leq \, x - 4\)
These inequalities don't just define a single line **(as in linear equations)** but a whole region on the coordinate plane. The solutions to these inequalities form a set of points (x, y) that can be plotted on a coordinate plane.
The key steps to graphing inequalities involve first converting the inequalities to equations, plotting the boundary line, and then shading the appropriate side of the line that represents the inequality.

A linear inequality looks much like a linear equation but with an inequality sign instead of an equals sign. Linear inequalities can be written in forms such as \(ax + by > c\), \(ax + by < c\), \(ax + by \leq c\), or \(ax + by \geq c\). For our example:
- The first inequality is \(y \, \geq \, \frac{1}{3} \ x - 2\). This is equivalent to \(y = \frac{1}{3} \ x - 2\) which can be plotted as a boundary line.
- The second inequality is \(y \, \leq \, x - 4\), similarly, we plot this as \(y = x - 4\) for the boundary.
The boundary line helps us visualize the limits beyond which the inequality does not hold. The inequality symbol will dictate which direction we should shade:

• Greater than (≥ or >): Shade above the line.
• Less than (≤ or <): Shade below the line.

By shading, we represent all the possible (x, y) values that satisfy the inequality.

The **solution set** for inequalities includes all the points that satisfy both inequalities in a system. To find this:
1. Plot the boundary lines for both inequalities on the same coordinate plane.
2. Shade the area that satisfies each inequality.
For our exercise, shading the area above \(y = \frac{1}{3} \ x - 2\) and below \(y = x - 4\) helps us visualize the solution set. The overlap of these regions represents the common solutions.
If these shaded regions overlap, this overlap is our solution set. If they do not overlap, the solution set is empty, indicating there are no common solutions that meet both inequalities.

Graphing on the coordinate plane allows us to visually represent our inequalities and easily identify the solution set. Here are the basic steps:
1. **Plot the boundary lines: Convert each inequality to an equation and plot it.** For \( y \, \geq \, \frac{1}{3} \ x - 2\) and \(y \, \leq \, x - 4\), plot the lines \( y = \, \frac{1}{3} \ x - 2\) and \( y = \, x - 4\).
2. **Decide on line style: Use a solid line for ≤ or ≥ and a dashed line for < or >.** Both our inequalities use solid lines.
3. **Shade the appropriate region:** To determine which side to shade, pick a test point (0,0) and see if it satisfies the inequality. Then shade accordingly.

• For \( y \, \geq \, \frac{1}{3} \ x - 2\), shade above the line.
• For \( y \, \leq \, x - 4\), shade below the line.

The region where both shades overlap is the solution set. In our example, this is the intersection of the two shaded areas. With practice, this method of graphing becomes intuitive and allows for quick visual representation of solutions.