Graphing Linear Inequalities on the Coordinate Plane: Solid vs. Dashed Lines and Shading Regions

When graphing y>2x+1 or y≥2x+1, the boundary line is the equation y=2x+1. The difference is whether the boundary is included in the solution. If the inequality is y≥2x+1 (greater than or equal), the boundary line is solid (included in the solution). If the inequality is y>2x+1 (greater than only), the boundary line is dashed (not included, excluded). Dashed means points on the line do not satisfy the inequality; solid means they do. This distinction determines whether points exactly on the line are solutions: solid line means yes, dashed means no.

After drawing the boundary line, shade the region where the inequality is true. For y>2x+1, shade above the line (where y is greater). For y<2x+1, shade below the line (where y is less). Test a point to verify: Pick (0,0). Is 0>2(0)+1? Is 0>1? No. So (0,0) is not in the solution region. Shade the region not containing (0,0), which is above the line y=2x+1.

To graph any linear inequality: First, identify the boundary line by writing the equality: y=2x+1. Second, determine if the boundary is solid (≤ or ≥) or dashed (< or >). Third, graph the boundary line on the coordinate plane. Fourth, pick a test point not on the line (usually (0,0) unless the line passes through it). Fifth, substitute the test point into the inequality. Sixth, if true, shade the region containing the test point; if false, shade the other region. This routine is foolproof: it works for all linear inequalities, and testing prevents shading the wrong region.

Example: Graph y<-x+3. Boundary: y=-x+3. Dashed line (< not ≤). Graph y=-x+3. Test (0,0): Is 0<-0+3? Is 0<3? Yes. Shade the region containing (0,0), which is below the line. This visual approach prevents errors that come from trying to remember "greater than means above" without understanding why.

Example 1: Graph x+y≥2. Boundary: x+y=2. Solid line (≥). Test (0,0): Is 0+0≥2? No. Shade the region not containing (0,0). Example 2: Graph y≤1/2 x. Boundary: y=1/2 x. Solid line (≤). Test (1,1): Is 1≤1/2(1)? Is 1≤1/2? No. Shade the region not containing (1,1).

Example 3: Graph -x+y>3. Rewrite as y>x+3. Boundary: y=x+3. Dashed line (>). Test (0,0): Is 0>0+3? No. Shade the region not containing (0,0). Students often shade the wrong region by remembering "less than=below" instead of testing. Testing takes 5 seconds and prevents wrong shading.

Graph ten inequalities per week on grid paper, using the step-by-step routine. For each, identify boundary type (solid/dashed), draw the line, test a point, and shade. After graphing, write the inequality from a shaded region shown on a graph (reverse the process). This two-way practice builds fluency: you can go from inequality to graph and from graph to inequality.

By week two, graphing becomes automatic. You recognize boundary types instantly and shade correctly without testing every time. On SAT questions about inequality graphs (rarely explicit graph questions, but sometimes implicit in problem-solving), you will know exactly which region is the solution and can evaluate whether specific points satisfy the inequality.