Linear inequalities are mathematical statements that compare two expressions using inequality symbols (, ≤, ≥). These inequalities represent a range of values rather than a single solution, unlike equations. The solution set of a linear inequality is the set of all points that satisfy the inequality....
Graphing Linear Inequalities
To graph a linear inequality, follow these steps:
- Rewrite the inequality in slope-intercept form (y = mx + b): This allows us to easily identify the slope (m) and y-intercept (b) of the line. In our example, 2x - y 2x - 4.
- Graph the boundary line: Plot the line represented by the equation y = 2x - 4. This line divides the coordinate plane into two regions. Since the inequality is 'y > 2x - 4', we use a dashed line to represent the boundary line, as the points on the line itself are not included in the solution set.
- Choose a test point: Select a point that does not lie on the boundary line. For example, we could choose (0, 0).
- Substitute the test point into the inequality: In this case, substituting (0, 0) into y > 2x - 4 gives us 0 > 2(0) - 4, which simplifies to 0 > -4. This statement is true.
- Shade the appropriate region: Since the test point (0, 0) satisfies the inequality, the region containing this point should be shaded. In our case, this would be the region above the dashed line.
Identifying the Correct Graph
The graph representing the solution set of the inequality 2x - y
- A dashed line representing the equation y = 2x - 4.
- The region above the line shaded, as this region contains the points that satisfy the inequality y > 2x - 4.
Example of a Correct Graph
Below is an example of a graph where the shaded region correctly represents the solution set for the inequality 2x - y
y
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(0, 4) |
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| /
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(0, 0) | /
| /
| /
|/
--------------------------> x
(2, 0)
Key Considerations
- Solid vs. dashed line: A solid line indicates that the points on the boundary line are included in the solution set. A dashed line indicates that the points on the boundary line are not included.
- Shading: The shaded region represents the set of all points that satisfy the inequality. The direction of the shading is determined by the direction of the inequality symbol. For 'greater than' inequalities, shade above the line. For 'less than' inequalities, shade below the line.
- Test point: The choice of the test point is arbitrary. Any point not on the boundary line will work.
Conclusion
By understanding the process of graphing linear inequalities, you can identify the correct graph representing the solution set of an inequality. Remember to consider the slope-intercept form of the inequality, the boundary line, the test point, and the direction of shading. This knowledge is crucial for solving and visualizing inequalities in various applications.