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Understanding linear inequalities is a fundamental aspect of learning algebra. A linear inequality looks much like a linear equation, except it uses inequality signs (\textless, \textgreater, \textless=, \textgreater=) instead of an equals sign. Instead of one solution, linear inequalities express a range of possible solutions. To solve them, we follow similar steps to solving a linear equation but with a key difference: if we ever multiply or divide by a negative number, we must flip the inequality sign.

For example, consider the inequality from the exercise, which can be written as: \[x - y \textless= 0\]. The solutions to this inequality are all the points \((x, y)\) on the Cartesian plane that make the inequality true. If you were to plot each solution as a point on a graph, you would end up shading an entire region that represents all possible solutions. This approach turns an abstract algebraic concept into a visual one, which many students find easier to comprehend.

Before you can graph the solution to a linear inequality, you must understand how to graph a linear equation. Graphing equations is a way to visualize how two variables are related to each other. Every linear equation can be graphed as a straight line that extends infinitely in both directions.

Here’s the basic process using the exercise as an example:

• Rewrite the linear inequality as an equation, like \(x - y = 0\).
• Solve for \(y\) to find the y-intercept and the slope, which in this case gives us \(y = x\), indicating a slope of 1 and a y-intercept at the origin (0,0).
• Plot the y-intercept and use the slope to find another point or plot multiple points that satisfy the equation.
• Draw a line through the points to represent all possible coordinates \((x, y)\) that solve the equation.

When graphing the inequality, instead of drawing a line, we're identifying a region. And whether the line is solid or dashed depends on whether the inequality includes equality (solid) or not (dashed).

Solving inequalities involves finding all the possible values that make the inequality true. This process can be done algebraically or graphically. Algebraically, the goal is to isolate the variable on one side of the inequality. Graphically, the solution to an inequality is represented by shading the area of the graph where the inequality holds.

To solve graphically, as in the exercise, follow these steps:

• Graph the boundary line, which is the related linear equation obtained by replacing the inequality symbol with an equals sign.
• Choose a test point that's not on the line and substitute it into the inequality.
• If the test point satisfies the inequality, then the region that includes the test point is the solution region.
• If the test point does not satisfy the inequality, then the opposite region is the solution region.
• Shade the region that is the solution to the inequality.

The importance of picking a test point cannot be overstressed—it's the crucial step that determines which side of the line represents the solution set. A common test point to use is (0,0) because it's easy to evaluate, except when your line passes through the origin.