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Linear inequalities are much like linear equations, but instead of simply finding where a line exists, we find where a range of solutions exists. A linear inequality looks like a linear equation but uses inequality symbols (<, >, ≤, ≥) instead of an equal sign. For example, in the inequality y > -1, we are expressing that y can be any real number greater than -1.

Understanding how to solve and graph these inequalities is essential for clearly visualizing the set of all possible solutions. It's not just about the line itself, but about the area that the line defines on one side or the other. In this way, a linear inequality divides the Cartesian plane into two halves, each of which may or may not be part of the solution set depending on the inequality's direction.

Graphing an inequality helps us visually understand all the values that satisfy the inequality. Unlike equations, where we typically draw solid lines to represent all points on the line as solutions, with inequalities we have a few more elements to consider. There are four key steps to graphing an inequality:

• Identify the inequality type: If the inequality sign is 'strict' (< or >), we use a dashed line to indicate that points on the line are not included. Conversely, if it's 'non-strict' (≤ or ≥), we use a solid line because points on the line are included.
• Draw the boundary line: This line represents where the inequality either begins or ends. It's the graph of the associated linear equation that would occur if the inequality sign was replaced with an equal sign.
• Shade the correct area: Pick a test point not on the boundary line. If the point satisfies the inequality, the area containing that test point is where all solutions lie and should be shaded. If it doesn't satisfy the inequality, shade the opposite side.
• Label your graph: It's good practice to label the line and the shaded area with the inequality it represents. This is especially handy when dealing with systems of inequalities.

The Cartesian plane is a fundamental concept in mathematics, named after René Descartes. It consists of two perpendicular number lines that intersect at their zero points, creating a grid we can use to plot points, lines, and curves using an ordered pair of numbers (x, y) which represent horizontal and vertical distances from the origin.

The plane is divided into four quadrants, each with a specific sign for the x and y values that the points contain. For graphing inequalities, the Cartesian plane acts as the stage where we visually represent the solutions to the inequality. Real-world applications of using a Cartesian plane can be seen in map plotting, computer graphics, and many other areas involving spatial analysis. By mastering graphing on this plane, students can create a visual representation of complex mathematical concepts.