Inequalities in Two Varıancs ality. 46. \( y<2 x+1 \)

To analyze the inequality \( y < 2x + 1 \), we can follow these steps: 1. **Identify the boundary line**: The inequality \( y < 2x + 1 \) can be rewritten as an equation to find the boundary line: \[ y = 2x + 1 \] This is a linear equation with a slope of 2 and a y-intercept of 1. 2. **Graph the boundary line**: Since the inequality is strict (i.e., \( < \)), we will draw a dashed line for \( y = 2x + 1 \) to indicate that points on the line are not included in the solution set. 3. **Determine the region of the inequality**: To find which side of the line satisfies the inequality \( y < 2x + 1 \), we can test a point that is not on the line. A common choice is the origin (0, 0). - Substitute \( x = 0 \) and \( y = 0 \) into the inequality: \[ 0 < 2(0) + 1 \implies 0 < 1 \] This statement is true, so the region that includes the origin is part of the solution set. 4. **Shade the appropriate region**: Since the inequality is \( y < 2x + 1 \), we will shade the area below the dashed line. ### Summary - The boundary line is \( y = 2x + 1 \) (dashed). - The solution set is the region below this line, including all points where \( y \) is less than \( 2x + 1 \). If you need a visual representation or further analysis, please let me know!