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The slope-intercept form is a way to express a linear equation using the formula \(y = mx + b\). In this form, \(m\) represents the slope of the line, and \(b\) is the y-intercept, which is where the line crosses the y-axis.
This format is particularly helpful for graphing lines easily, as you can begin plotting the line by starting at the y-intercept and then using the slope to find additional points.
The slope \(m\) indicates the steepness and direction of the line - a positive slope means the line goes upwards as you move from left to right, whereas a negative slope means it goes downwards. The y-intercept \(b\) gives the starting point of the line on the y-axis, making it easy to plot.

Graphing linear inequalities involves plotting not just a line but a region of the graph. The inequality symbol (>, <, ≥, ≤) indicates which side of a particular line is included in the region.
For instance, in the example inequality \(y > \frac{2}{5}x - 4\), the graph includes not just the line itself, but the half-plane above it. To graph an inequality:

• Start by graphing the associated equation without the inequality (making the inequality an equality).
• Determine whether the line should be solid or dashed. Dashed lines are used for \(>\) or \(<\) as they don't include the boundary line itself, while solid lines are for \(\geq\) or \(\leq\) as they do.

This will allow you to focus on which area forms the solution region.

Boundary lines are essential in defining linear inequalities. They help specify the limits of the solution area on a graph. The boundary line is derived from the associated linear equation \(y = mx + b\) where the inequality sign is treated as an equal sign.
For \(y = \frac{2}{5}x - 4\), this line serves as the divider of the coordinate plane. Whether this line is solid or dashed depends on the type of inequality:

• Use a solid line if the inequality is \(\leq\) or \(\geq\), which includes the boundary itself as part of the solution.
• Use a dashed line if the inequality is \(<\) or \(>\), meaning the boundary is not part of the solution.

With this understanding, you can correctly mark the boundary on a graph.

Shading the solution region is a crucial step in graphing inequalities and helps convey the set of points that satisfy the inequality.
After drawing the boundary line, you need to determine which side of this line contains solutions. A simple method is to pick a test point that isn’t on the line, typically the origin (0,0) if possible, and substitute it into the inequality.
If the test point satisfies the inequality, then the region containing this test point is shaded, as it includes all the solutions of the inequality.

• If the test point does not satisfy the inequality, shade the opposite side of the line.

In the example \(y > \frac{2}{5}x - 4\), since (0,0) satisfies the inequality, you would shade above the dashed line, indicating all points that are solutions.