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Inequalities represent a range of values rather than a specific value. When dealing with inequalities in the context of graphing, this means that the solution is not just a line but a region on the graph. For example, the inequality \(5x - 3y \geq 15\) indicates a set of points where the combination of \(x\) and \(y\) values satisfies the inequality.

Inequalities use symbols such as:

• \(>\) means greater than.
• \(<\) means less than.
• \(\geq\) means greater than or equal to.
• \(\leq\) means less than or equal to.

When graphing inequalities, a solid line is used when the inequality includes equal to (like \(\geq\) or \(\leq\)), showing that points on the line itself satisfy the inequality. For strict inequalities (like \(>\) or \(<\)), a dashed line is used to indicate that points on the line are not part of the solution.

Graphing a linear equation involves plotting points that lie on the line represented by the equation. Linear equations are typically in the form \(Ax + By = C\), but we often convert them into a format that's more convenient for graphing, called the slope-intercept form. By plotting the equation \(5x - 3y = 15\) as a line, we lay the groundwork for solving the inequality \(5x - 3y \geq 15\).
Converting to slope-intercept form (\(y = mx + b\)) makes it easier to identify key features of the line:

• The slope \(m\), which describes the angle and direction of the line.
• The y-intercept \(b\), where the line crosses the y-axis.

In our example, the equation becomes \(y = \frac{5}{3}x - 5\), helping us locate the y-intercept at \((0, -5)\) and use the slope \(\frac{5}{3}\) to find other points.

Slope-intercept form \(y = mx + b\) is a handy equation style that simplifies graphing linear equations by making the slope and y-intercept easily recognizable. Here, \(m\) stands for the slope, and \(b\) is the y-intercept.

• The slope \(m\) shows how much \(y\) increases when \(x\) increases by 1 unit. A positive slope means the line rises, and a negative slope means it falls.
• The y-intercept \(b\) tells us where the line crosses the y-axis, setting the starting point for the graph.

In the example \(y = \frac{5}{3}x - 5\), we use these values to draw the graph on a Cartesian plane, plotting the y-intercept first and using the slope to determine the line's incline. This approach is fundamental for visualizing and solving inequalities graphically.

When graphing inequalities, correctly identifying the shaded region is crucial, as it visually represents the set of solutions to the inequality. After plotting the line, the next step is to determine which side of the line contains the solutions.
The inequality \(5x - 3y \geq 15\) tells us that the solutions lie on or above the line \(y = \frac{5}{3}x - 5\).

• If the inequality sign points upwards (\(\geq\) or \(>\)), shade above the line.
• If it points downwards (\(\leq\) or \(<\)), shade below the line.

Remember to check whether the line should be included in the solution by using a solid line for \(\geq\) or \(\leq\) inequalities. This approach helps convey the correct range of possible values for both \(x\) and \(y\) that satisfy the inequality, effectively solving the problem.