In Exercises 1–26, graph each inequality. y>2x−1

Verified step by step guidance

1. Step 1: Start by identifying the boundary line for the inequality. The inequality given is y > 2x - 1. The boundary line is y = 2x - 1.
2. Step 2: Graph the boundary line y = 2x - 1. Since the inequality is 'greater than' and not 'greater than or equal to', use a dashed line to indicate that points on the line are not included in the solution.
3. Step 3: Choose a test point that is not on the boundary line to determine which side of the line to shade. A common test point is (0,0).
4. Step 4: Substitute the test point (0,0) into the inequality y > 2x - 1. If the inequality holds true, shade the region that includes the test point. If not, shade the opposite side.
5. Step 5: Since the inequality is y > 2x - 1, shade the region above the dashed line, as this represents all the points where y is greater than 2x - 1.

Key Concepts

Inequalities

Inequalities are mathematical expressions that show the relationship between two values when they are not equal. They use symbols such as '>', '<', '≥', and '≤' to indicate whether one side is greater than, less than, or equal to the other. Understanding how to interpret and manipulate inequalities is essential for solving problems that involve ranges of values.

Graphing Linear Equations

Graphing linear equations involves plotting points on a coordinate plane that satisfy the equation. The equation y = mx + b represents a line where 'm' is the slope and 'b' is the y-intercept. For inequalities, the graph will include a boundary line and shading to indicate the solution set, which represents all the points that satisfy the inequality.

Shading Regions in Graphs

When graphing inequalities, it is important to shade the appropriate region of the graph to represent all possible solutions. For example, in the inequality y > 2x - 1, the area above the line y = 2x - 1 is shaded, indicating that all points in this region satisfy the inequality. Understanding how to determine which side of the line to shade is crucial for accurately representing the solution.