How To Graph Inequalities

Graphing Inequalities: A Comprehensive Guide

Inequalities are mathematical statements that express an unequal relationship between two expressions. They are commonly used to represent constraints or limitations in real-world problems. Graphing inequalities allows us to visualize the solution set and understand the relationships between the variables involved.

Types of Inequalities

There are two main types of inequalities:

1. Linear inequalities: These inequalities can be expressed in the form of ax + b < c, ax + b > c, ax + b ≤ c, or ax + b ≥ c, where a, b, and c are constants and x is the variable.
2. Absolute value inequalities: These inequalities involve the absolute value function and can be expressed in the form of |x - a| < b, |x - a| > b, |x - a| ≤ b, or |x - a| ≥ b, where a and b are constants and x is the variable.

Graphing Linear Inequalities

To graph linear inequalities, follow these steps:

1. Plot the boundary line: Find the equation of the boundary line by setting the inequality equal to zero and solving for y. This line divides the plane into two half-planes.
2. Determine the shading: If the inequality is less than (< or ≤), shade the half-plane below the boundary line. If the inequality is greater than (> or ≥), shade the half-plane above the boundary line.
3. Test a point: Choose a point that is not on the boundary line and substitute it into the inequality. If the inequality is true, shade the half-plane that contains that point. If it is false, shade the other half-plane.

Graphing Absolute Value Inequalities

To graph absolute value inequalities, follow these steps:

1. Plot the center line: Find the center line of the inequality by setting the absolute value expression equal to zero and solving for x. This line represents the boundary between the two possible solutions.
2. Determine the direction of the inequality: If the inequality is less than (< or ≤), shade the region outside the two vertical lines parallel to the center line and a distance of b away. If the inequality is greater than (> or ≥), shade the region inside the two vertical lines.
3. Test a point: Choose a point that is not on the boundary line and substitute it into the inequality. If the inequality is true, shade the region that contains that point. If it is false, shade the other region.

Examples

Example 1: Graph the linear inequality 2x + 1 < 5.

1. Plot the boundary line: y = -2x + 4.
2. Determine the shading: Shade the half-plane below the boundary line.
3. Test a point: (0, 0): 2(0) + 1 < 5 (true). Shade the half-plane that contains (0, 0).

Example 2: Graph the absolute value inequality |x – 2| > 3.

1. Plot the center line: x = 2.
2. Determine the direction of the inequality: Shade the region outside the vertical lines x = -1 and x = 5.
3. Test a point: (0, 0): |0 – 2| > 3 (true). Shade the region that contains (0, 0).

FAQ

1. What is the difference between an equality and an inequality?

An equality (=) expresses that two expressions are equal, while an inequality (<, >, ≤, or ≥) expresses that two expressions are unequal.

2. What is the solution set of an inequality?

The solution set of an inequality is the set of all values that make the inequality true.

3. How do I determine the direction of the inequality when graphing an absolute value inequality?

If the inequality is less than (< or ≤), shade the region outside the vertical lines. If the inequality is greater than (> or ≥), shade the region inside the vertical lines.

4. What if the inequality is an equation?

If the inequality is an equation (=), graph it as a solid line and shade only the half-plane or region that satisfies the inequality.

5. How can I check if a point satisfies an inequality?

Substitute the point into the inequality and see if it makes the inequality true or false.

Conclusion

Graphing inequalities is a fundamental skill in mathematics, allowing us to visualize and solve problems involving constraints. By understanding the different types of inequalities and their graphing techniques, we can effectively represent and interpret mathematical relationships.