BackGraphing the Intersection of Two Linear Inequalities 2.4
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Graphing the Intersection of Two Linear Inequalities
Introduction to Linear Inequalities
Linear inequalities are mathematical statements that compare linear expressions using inequality symbols such as <, >, ≤, or ≥. The solution to a linear inequality is a set of points in the plane that satisfy the inequality.
Linear Inequality: An expression of the form ax + by < c, ax + by > c, ax + by ≤ c, or ax + by ≥ c.
Solution Set: The set of all ordered pairs (x, y) that make the inequality true.
Intersection of Two Linear Inequalities
The intersection of two or more linear inequalities is the region of the plane where all points satisfy all of the inequalities simultaneously. This is often referred to as the solution set of the system of inequalities.
Intersection: The overlapping region that satisfies every inequality in the system.
"And" Statement: When inequalities are joined by "and," only points that satisfy all inequalities are included in the solution set.
Example: Graphing the Intersection
Consider the system:
To graph the intersection:
Rewrite the inequalities in slope-intercept form:
Subtract from both sides of the first inequality:
The second inequality is already in slope-intercept form:
Graph each inequality:
Graph the line as a dashed line (since the inequality is strict, >).
Shade the region above the line for .
Graph the line as a dashed line (since the inequality is strict, >).
Shade the region above the line for .
Find the intersection:
The solution set is the region where the shaded areas for both inequalities overlap.
Only points in the darkest shaded region satisfy both inequalities.
Verifying Solutions
To verify that a point is in the solution set, substitute its coordinates into both inequalities. If both are true, the point is in the intersection.
Example: Test the point (2, 4):
First inequality: (True)
Second inequality: (True)
Therefore, (2, 4) is in the solution set.
Summary Table: Steps for Graphing Intersection of Linear Inequalities
Step | Description |
|---|---|
1 | Rewrite each inequality in slope-intercept form () |
2 | Graph each boundary line (dashed for < or >, solid for ≤ or ≥) |
3 | Shade the region representing the solution to each inequality |
4 | Identify the overlapping (intersection) region |
5 | Verify points in the intersection by substitution |
Additional info:
In systems with more than two inequalities, the intersection is the region where all shaded areas overlap.
For inequalities with "≤" or "≥", use solid lines to indicate that points on the line are included in the solution set.
