3.6 Inequalities on Graphs
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Learning Objectives
- Understand how inequalities can be represented graphically
- Identify regions that satisfy linear and quadratic inequalities
- Interpret shaded regions on coordinate planes
- Use graphing techniques to solve systems of inequalities
- Apply graphical inequalities in real-world problem-solving
Key Concepts
Introduction to Graphical Inequalities
Graphical inequalities show the set of points (x, y) that satisfy an inequality.
Instead of solving algebraically, we represent the solution as a shaded region on the coordinate plane.
Inequalities can involve straight lines (linear), curves (quadratic), or multiple lines in systems.
Graphing Linear Inequalities
Linear inequalities are of the form ax + by < c, ax + by > c, ax + by ≤ c, or ax + by ≥ c.
Step 1: Rearrange to the form y = mx + c if needed.
Step 2: Plot the boundary line (use a dashed line for < or >, solid for ≤ or ≥).
Step 3: Shade the correct side based on whether y is greater or less than the expression.
Example 1
Graph the inequality: y < 2x + 1
Solution:
Step 1: Plot the line y = 2x + 1 with a dashed line (since it's <). Step 2: Shade the region **below** the line, because we want y-values less than 2x + 1.
Example 2
Graph the inequality: y ≥ -x + 4
Solution:
Step 1: Plot the line y = -x + 4 with a solid line (since it's ≥). Step 2: Shade the region **above** the line.
Solving Quadratic Inequalities on Graphs
When graphing a quadratic inequality such as y ≤ x² - 4, the region is defined by the curve and the area above or below it.
Step 1: Plot the quadratic curve (e.g. y = x² - 4).
Step 2: Use a solid curve for ≤ or ≥, and a dashed one for < or >.
Step 3: Shade the region that satisfies the inequality (typically below the curve for ≤).
Example 1
Graph the inequality: y ≥ x² - 2x - 3
Solution:
Plot the curve y = x² - 2x - 3 with a solid line. Shade the region **above** the parabola since y is greater than or equal to the expression.
Systems of Inequalities
You can solve systems of inequalities by graphing multiple inequalities on the same axes.
The solution is the overlapping shaded region that satisfies all inequalities simultaneously.
Example 1
Graph the system: y > x + 1 and y < 3x - 2
Solution:
Plot both lines using dashed lines. Shade **above** y = x + 1 and **below** y = 3x - 2. The solution is the region between the two lines where the shadings overlap.
Applications and Interpretation
Inequalities on graphs are used in optimization problems, economics, and feasibility regions in linear programming.
Graphical representations help visualize constraints and identify viable solutions.
Test Your Knowledge
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