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Graphing inequalities is a fundamental skill in algebra that combines an understanding of coordinate systems with the properties of inequalities. Instead of just drawing a line like you would for a linear equation, graphing an inequality illustrates a whole region of the graph. To graph an inequality like \( -3x + y \leq 2 \), we first treat it as an equation (\( -3x + y = 2 \) in this case) to find the boundary line.

Once the boundary line is drawn, which is done by plotting points such as intercepts, the next step is to determine which side of the line represents the solution set. If the inequality symbol includes an equal sign (\leq), the boundary line is solid to show that points on the line are part of the solution. With strict inequalities (\(<\) or \(>\)), the line would be dashed, indicating that the points on the line are not part of the solution set. After identifying the correct side using a test point, the area representing the solution is shaded on the graph. This shaded region includes all points that make the inequality true.

Intercepts are where the graph of an equation crosses the axes. The x-intercept is the point where the graph crosses the x-axis and the y-intercept is where it crosses the y-axis. To find these, we set the opposite variable to zero and solve for the remaining one.

For the given equation \( -3x + y = 2 \), setting \( x = 0 \) gives the y-intercept as \( y = 2 \), and setting \( y = 0 \) provides the x-intercept \( x = \frac{-2}{3} \). This is a crucial step because these intercepts are used to draw the boundary line of the inequality on the graph. They serve as reference points and are important for understanding the behavior of the equation along the axes.

Linear equations form the basis of algebra and graphing. They represent lines on a coordinate grid and have a standard form of \( Ax + By = C \). In a linear equation, \( A \), \( B \) and \( C \) are constants, and \( x \) and \( y \) are variables.

When graphing a linear equation like \( -3x + y = 2 \), we are essentially finding all x and y pairs that make the equation true. This set of pairs form a straight line. If \( B eq 0 \) (which is our case since \( B = 1 \) in \( -3x + y \)), we can solve for \( y \) to get the slope-intercept form \( y = mx + b \), where \( m \) stands for the slope and \( b \) stands for the y-intercept. Linear equations can model real-world situations and understanding how to work with them is an essential skill for students.

Inequality shading is the process used to show the solution set of an inequality on a graph. After the boundary line defined by the related linear equation is drawn, shading is applied to one side of the line to represent all the possible solutions. To determine which side to shade, we use a test point that is not on the line.

For an inequality like \( -3x + y \leq 2 \), after drawing the solid line for the boundary, we might choose the origin (0,0) as our test point. Substituting this into the inequality \( -3(0) + 0 \leq 2 \) confirms that the origin makes the inequality true, so we shade the side containing (0,0). In this instance, all points in the shaded area, including those on the boundary line, satisfy the inequality. This visual representation of the solution set is an invaluable tool in understanding and solving inequalities.