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The slope-intercept form of a linear equation or inequality is a way to express lines in a simple format: \(y = mx + c\). Here, \(m\) represents the slope of the line, while \(c\) is the y-intercept, the point where the line crosses the y-axis. This format is especially useful when quickly graphing a line. To convert any linear equation to this form, solve for \(y\) in terms of \(x\).
In our example, the inequality \(2x - y > 4\) is first rearranged to \(-y > -2x + 4\), and then multiplied by -1 (remembering to flip the inequality sign) to become \(y < 2x - 4\). Now, it's in slope-intercept form, allowing us to see at a glance that the slope \(m\) is 2 and the y-intercept \(c\) is -4. This tells us that the line rises by 2 units for each 1 unit it moves to the right, and starts at \(-4\) on the y-axis.

Linear inequalities, much like linear equations, involve expressions of the first degree, but with inequality signs (">", "<", ">=", "<=") instead of an equals sign. These inequalities describe a collection of points, forming a region on a graph, rather than a single line. They express relationships where one side of the equation may be greater or less than the other, instead of equal.
When graphing linear inequalities, it's essential to adjust the line style based on the inequality sign. For instance, "<" or ">" result in a dashed line, indicating points on the line do not satisfy the inequality. Conversely, "≤" or "≥" result in a solid line, where points on the line are included in the solution set. In the exercise, \(y < 2x - 4\) uses a dashed line since it excludes the actual line as part of the solution.

The solution region of a linear inequality consists of all the points that satisfy the inequality. Once you have graphed the boundary line (dashed or solid), you must determine which side of this line represents the solution region.
For the inequality \(y < 2x - 4\), the region lies below the line because we are looking for points where \(y\) is less than \(2x - 4\). To find out the correct side to shade, you can pick a test point not on the line, like (0,0), and substitute it into the inequality. If it holds true, that's your shaded region! In conclusion, all the points within this shaded area are solutions to the inequality.

• Graphing helps visualize solutions.
• Dashed lines exclude boundary points.
• Shaded regions signify all solutions.