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Graphing an inequality involves plotting a boundary line on the coordinate plane and shading a region to represent all the possible solutions that satisfy the inequality. For example, let's consider the inequality \(x \leq 2\). To graph this inequality, you would first draw the vertical line \(x = 2\), which would serve as the boundary.

The inequality indicates that we're interested in all the values less than or equal to 2; therefore, the region to the left of this vertical line (including the line itself, since it's \(\leq\) and not \(<\)) is shaded. Similarly, for the inequality \(y \geq -1\), the horizontal line \(y = -1\) is drawn, and the region above this line is shaded because \(y\) values are larger than or equal to -1.

Graphing inequalities requires attention to the type of line used for the boundary as well. A solid line is used to represent \(\leq\) or \(\geq\) since the points on the line are included in the solutions, while a dashed line is used for \(<\) or \(>\) since the points on the line are not included.

The solution set of a system of inequalities is the region of the plane where the shaded areas of all individual inequalities overlap. In essence, it is the set of all possible points \((x, y)\) that satisfy all the inequalities in the system simultaneously. Referencing our exercise, the solution set is found at the intersection of the shaded regions for each inequality.

The solution set can be visualized as the common shaded region. For the system including \(x \leq 2\) and \(y \geq -1\), the solution set is the area to the left of or on the line \(x = 2\) and above or on the line \(y = -1\). When graphed, it is the region that is doubly shaded and represents the set of coordinates that adhere to both conditions at once. It's often helpful to test points within the shaded region to verify that they meet all constraints of the inequalities.

Graphing linear inequalities is similar to graphing linear equations, but instead of a line representing all the solutions, a half-plane is identified as the solution set. For each linear inequality, start by graphing the corresponding linear equation as if the inequality symbol were an equal sign.

For instance, with the inequality \(y \geq -1\), graph the line \(y = -1\) first. Since this is an inequality rather than an equation, the next step is to choose which side of the line represents the solution set. If the inequality is greater than (\(>\)) or greater than or equal to (\(\geq\)), you shade above the line. Conversely, if the inequality is less than (\(<\)) or less than or equal to (\(\leq\)), you shade below the line.

Identifying Solid or Dashed Lines

Remember, use a solid line for inequalities including \(\leq\) or \(\geq\), indicating that points on the line satisfy the inequality. Use a dashed line for strict inequalities, \(<\) or \(>\), where the points on the line are not part of the solution set.

The key to graphing a system of linear inequalities is to individually plot and shade each inequality and then determine the overlapped region where both conditions are true.