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Let's start with the most basic concept of linear equations. A **linear equation** is an equation that makes a straight line when it is graphed on a coordinate plane. Linear equations generally have the form: - **y = mx + b** A **linear equation** represents a relationship between two variables, usually x and y, where changing one variable impacts the other. The constant *m* represents the slope (how steep the line is), and *b* represents the y-intercept (where the line crosses the y-axis). Even though our original problem involves inequalities, understanding linear equations is vital because we first rewrite the inequality as an equation: - **y = -x** Here, the slope *m* is -1, and the y-intercept *b* is 0.

A **coordinate plane** is a two-dimensional surface on which we can plot points, lines, and curves. It's defined by two perpendicular number lines, the x-axis (horizontal) and the y-axis (vertical), that intersect at a point called the origin (0,0). To graph any equation or inequality, we need to understand how to use the coordinate plane effectively. For example: - **Plotting Points**: Each point on the coordinate plane is represented by an ordered pair (x, y). For example, the point (3, 2) is located 3 units to the right of the origin on the x-axis and 2 units up on the y-axis. - **Quadrants**: The plane is divided into four quadrants. In the upper right quadrant, both x and y are positive. This is quadrant I. Moving counterclockwise, each quadrant changes in the sign of x and y values. In our exercise, we plot points such as (0,0) and (1,-1) to help draw the line y = -x on the coordinate plane. Understanding where to place these points is crucial to drawing the line accurately.

Simply put, **inequalities** show a relationship where two expressions are not equal. They use symbols like: <, >, ≤, and ≥. In our exercise, we deal with the inequality: **y ≤ -x** Here's what the symbols mean: - **<**: less than - **>**: greater than - **≤**: less than or equal to - **≥**: greater than or equal to When graphing inequalities: - **Solid Line**: Represents ≤ or ≥ as it shows that the points on the line are included in the solution. - **Dashed Line**: Represents < or > indicating that the points on the line are not part of the solution. In our case, **y ≤ -x** uses a solid line because the points on the line satisfy the inequality. We then shade the region below the line, as all those points satisfy y being less than or equal to -x.

Lastly, let's discuss **graphing**. This is the process of representing numerical information visually. For linear inequalities: - **Rewrite as an Equation**: Replace the inequality symbol with an equals sign to make it easier to plot initially: y = -x. - **Plot Key Points**: Find points that satisfy this equation. For example, since when x = 0, y = 0, plot (0,0). When x = 1, y = -1, plot (1,-1). - **Draw the Line**: Connect these points with a straight line. Remember, it’s solid for ≤ or ≥ inequalities. - **Shading the Region**: Determine which side of the line satisfies the inequality. For y ≤ -x, shade below the line, indicating the area where y values are less than or equal to -x. A quick way to check if shaded correctly is to use a test point. By following these steps, you can graph any linear inequality accurately. Understanding each part—from plotting points to shading regions—is essential for mastering graphing techniques.