91Ó°ÊÓ

The **boundary line** is the key to starting your graphing process for inequalities. It helps us understand where the solutions might lie.
Let's break down the steps to find the boundary line using the example inequality, \(3x - 4y \leq 12\).
First, change the inequality into an equation: \(3x - 4y = 12\).
Next, find the intercepts, which are points where the line crosses the x-axis and y-axis. For the x-intercept, set \(y = 0\) and solve for \(x\):
\[3x = 12 \rightarrow x = 4\].
For the y-intercept, set \(x = 0\) and solve for \(y\):
\[ -4y = 12 \rightarrow y = -3\].
Now, plot these intercepts: (4, 0) and (0, -3). Draw a straight line through these points. This line is the boundary line for the inequality.

After plotting the boundary line, the next important step is **shading the solution regions**. This tells us where the solutions to the inequality lie.
Consider the boundary line we drew for \(3x - 4y = 12\). Since the inequality is \(\leq\), we shade below the line.
For the second inequality \( x + y \geq -3\), convert it to the equation \(x + y = -3\) and draw its boundary line by finding the intercepts.
If \(x = 0\), \(y = -3\), and if \(y = 0\), \(x = -3\).
Connect these points to draw the line.
Now, shade above the line as the inequality is \(\geq\).
The solution set is where these shaded regions overlap. It helps to use different colors or patterns for each region and then indicate the overlapping area clearly.

Understanding **intercepts** is crucial to properly graph a line on a coordinate plane. Intercepts are points where the line crosses the x-axis and y-axis.
For the inequality system \(3x - 4y \leq 12\):
Exploring x-intercepts: Set \(y = 0\) and solve for \(x\):
\[3x = 12 \rightarrow x = 4\].
Thus, the x-intercept is (4, 0).
Exploring y-intercepts: Set \(x = 0\) and solve for \(y\):
\[ -4y = 12 \rightarrow y = -3\].
Thus, the y-intercept is (0, -3).
For the second inequality \(x + y \geq -3\):
Exploring x-intercepts: Set \(y = 0\) and solve for \(x\): \[x = -3\].
Thus, the x-intercept is (-3, 0).
Exploring y-intercepts: Set \(x = 0\) and solve for \(y\): \[y = -3\].
Thus, the y-intercept is (0, -3).
These intercepts are key points to find and connect to draw the boundary lines for each of the inequalities. They make it easy to visualize the equations on a grid.

**Inequality graphing** combines understanding of boundary lines, shading, and intercepts to visualize solutions to inequalities.
Start by converting inequalities into boundary line equations.
Plot the intercepts to draw lines. For example, convert \(3x - 4y \leq 12\) to \(3x - 4y = 12\), and \(x + y \geq -3\) to \(x + y = -3\).
Plot their intercepts and draw lines.
For \(3x - 4y\leq12\), shade below the line since the inequality sign is \( \leq\).
For \(x + y \geq-3\), shade above the line because the sign is \(\geq\).
The solution set is the overlapping shaded area.
It's often helpful to use colored pencils or different shading patterns to keep track.
Remember:

• For \(\leq\) or \(\geq\), use a solid line.
• For \(<\) or \(>\), use a dashed line.

The graph of a system of inequalities shows the region that satisfies all conditions. This visual representation makes solving complex inequality systems more intuitive.