Shading Rules for Slope-Intercept Form
When a linear inequality is in slope-intercept form, the shading direction follows a simple rule taught in Grade 11 enVision Algebra 1 (Chapter 4: Systems of Linear Equations and Inequalities): inequalities equivalent to y ≥ mx + b shade above the boundary line, and inequalities equivalent to y ≤ mx + b shade below. Students first convert the inequality to slope-intercept form by solving for y, then apply this rule. The boundary line is solid for ≤ or ≥ and dashed for strict inequalities.
Key Concepts
When a linear inequality is written in slope intercept form, the shading direction follows a simple rule. If the inequality is equivalent to $y \geq mx + b$, shade the half plane above the line. If it is equivalent to $y \leq mx + b$, shade below the line.
Common Questions
How do you determine which side to shade for a linear inequality?
Rewrite the inequality in slope-intercept form (solve for y). If y ≥ mx + b, shade above the line. If y ≤ mx + b, shade below.
When is the boundary line solid vs. dashed?
Use a solid line for ≤ or ≥ (the boundary is included). Use a dashed line for < or > (the boundary is not included).
Why do you need to solve for y before applying the shading rule?
The rule 'greater than shades above' only applies reliably when the inequality is in the form y compared to mx + b. Without isolating y first, the inequality sign may have been reversed during solving.
What happens to the inequality sign when you divide by a negative number?
The inequality sign flips direction. This is why careful attention is needed when solving for y.
How do you check whether the correct region is shaded?
Pick a test point not on the boundary line (often (0, 0)) and check whether it satisfies the original inequality. If it does, shade the region containing it.
What does the shaded region represent?
The shaded region contains all ordered pairs (x, y) that satisfy the inequality — all solutions to the linear inequality.