Graph each inequality. y>2x−1

Verified step by step guidance

1. Identify the inequality given: \(y > 2x - 1\). This represents all points \((x, y)\) where the \(y\)-value is greater than \(2x - 1\).
2. First, graph the boundary line \(y = 2x - 1\). This is a straight line with slope \(2\) and \(y\)-intercept \(-1\).
3. Since the inequality is strict (\(>\), not \(\geq\)), draw the boundary line as a dashed line to indicate points on the line are not included in the solution.
4. Choose a test point not on the line, commonly \((0,0)\), and substitute into the inequality: check if \(0 > 2(0) - 1\) which simplifies to \(0 > -1\). Since this is true, shade the region of the graph that contains \((0,0)\).
5. The shaded region represents all solutions to the inequality \(y > 2x - 1\). This completes the graph of the inequality.

Key Concepts

Graphing Linear Inequalities

Graphing linear inequalities involves first graphing the related linear equation as a boundary line. The inequality symbol determines whether the boundary is solid (for ≤ or ≥) or dashed (for < or >). The solution region is the set of points that satisfy the inequality, typically shaded on one side of the boundary line.

Slope-Intercept Form of a Line

The slope-intercept form, y = mx + b, expresses a line where m is the slope and b is the y-intercept. It helps quickly graph the line by starting at (0, b) and using the slope to find other points. For y > 2x - 1, the slope is 2 and the y-intercept is -1.

Testing Points to Determine the Solution Region

After graphing the boundary line, select a test point not on the line (often (0,0)) to check if it satisfies the inequality. If the test point makes the inequality true, shade the region containing that point; otherwise, shade the opposite side. This confirms the correct solution area.