Solving and Graphing Linear Inequalities
Linear inequalities are mathematical expressions similar to linear equations, but they use inequality symbols instead of an equal sign. This page covers how to identify solutions and graph linear inequalities.
Identifying Solutions
A solution to a linear inequality is any ordered pair that makes the inequality true. To determine if an ordered pair is a solution, substitute the values into the inequality and check if it holds true.
Example: For the inequality y < x - 1, we can check if (7, 3) is a solution:
3 < 7 - 1
3 < 6
This is true, so (7, 3) is a solution.
Example: For the inequality y > 3x + 2, we can check if (4, 5) is a solution:
5 > 3(4) + 2
5 > 14
This is false, so (4, 5) is not a solution.
Graphing Inequalities
When graphing linear inequalities, the type of line and shading used depends on the inequality symbol:
Highlight:
- For y ≤ or y ≥, use a solid line.
- For y < or y >, use a dashed line.
- For > or ≥, shade above the line.
- For < or ≤, shade below the line.
Vocabulary:
- Solid line: A continuous, unbroken line used for inequalities that include the boundary (≤ or ≥).
- Dashed line: A line made up of short, disconnected segments used for strict inequalities (< or >).
Understanding these rules is crucial for graphing linear inequalities step by step. When using a graphing linear inequalities calculator, it's important to input the inequality correctly to ensure accurate shading and line type.
Definition: A system of linear inequalities consists of two or more linear inequalities that must be satisfied simultaneously. The solution is the region where all inequalities are true.
Practicing with a graphing linear inequalities worksheet can help reinforce these concepts and improve your skills in solving linear inequalities with two variables. Remember, the solution of an inequality is not just a single point, but an entire region on the coordinate plane that satisfies the given conditions.
