Solving Linear
Solving Linear Inequalities
Inequalities in
in
Two Variables
Two Variables
Key Definitions & Concepts
Linear Inequality in Two Variables: An inequality that can be written in one of the
following forms:
≤ ≥ ≤
y<mx+b, y mx+b, y>mx+b, y mx+by<mx+b ,y mx+b, y>mx+b, y mx+b ≥
Boundary Line: The line graphed from the related equation y=mx+by=mx+b. It
separates the coordinate plane into two regions (half-planes).
Half-Plane: The region of the coordinate grid on one side of the boundary line that
contains all the ordered pairs (x,y)(x,y) that make the inequality statement true.
Solid vs. Dashed Boundary Line:
Dashed Line (<< or >>): Points on the boundary line are NOT included in the
solution set.
Solid Line (≤≤ ≥≥ or ): Points on the boundary line ARE included in the solution
set.
Test Point: An ordered pair—such as (0,0)(0,0)—substituted into the original
inequality to determine which half-plane to shade.
How toGraph & Solve Two-Variable Inequalities
Follow these 4 steps to graph the solution set of any two-variable linear inequality:
1. Identify & Graph the Boundary Line:
Treat the inequality as an equation (y=mx+by=mx+b).
Find the yy-intercept (bb) and use the slope (mm) to plot points.
2. Choose Line Type:
Draw a Dashed Line for strict inequalities (<< or >>).
≤≤ ≥≥
Draw a Solid Line for inclusive inequalities ( or ).
Solving Linear Inequalities
Inequalities in
in
Two Variables
Two Variables
Key Definitions & Concepts
Linear Inequality in Two Variables: An inequality that can be written in one of the
following forms:
≤ ≥ ≤
y<mx+b, y mx+b, y>mx+b, y mx+by<mx+b ,y mx+b, y>mx+b, y mx+b ≥
Boundary Line: The line graphed from the related equation y=mx+by=mx+b. It
separates the coordinate plane into two regions (half-planes).
Half-Plane: The region of the coordinate grid on one side of the boundary line that
contains all the ordered pairs (x,y)(x,y) that make the inequality statement true.
Solid vs. Dashed Boundary Line:
Dashed Line (<< or >>): Points on the boundary line are NOT included in the
solution set.
Solid Line (≤≤ ≥≥ or ): Points on the boundary line ARE included in the solution
set.
Test Point: An ordered pair—such as (0,0)(0,0)—substituted into the original
inequality to determine which half-plane to shade.
How toGraph & Solve Two-Variable Inequalities
Follow these 4 steps to graph the solution set of any two-variable linear inequality:
1. Identify & Graph the Boundary Line:
Treat the inequality as an equation (y=mx+by=mx+b).
Find the yy-intercept (bb) and use the slope (mm) to plot points.
2. Choose Line Type:
Draw a Dashed Line for strict inequalities (<< or >>).
≤≤ ≥≥
Draw a Solid Line for inclusive inequalities ( or ).
