13. \( y \leq-\frac{3}{4} x-1 \)

The inequality \( y \leq -\frac{3}{4} x - 1 \) represents a region in the coordinate plane. To understand this inequality better, we can follow these steps: 1. **Identify the boundary line**: The equation of the line is given by \( y = -\frac{3}{4} x - 1 \). This line has a slope of \(-\frac{3}{4}\) and a y-intercept of \(-1\). 2. **Graph the boundary line**: - Start at the y-intercept (0, -1). - From there, use the slope to find another point. The slope \(-\frac{3}{4}\) means that for every 4 units you move to the right (positive x-direction), you move down 3 units (negative y-direction). - From (0, -1), moving 4 units right to (4, -1) and then down 3 units gives you the point (4, -4). - Plot these points and draw a straight line through them. 3. **Determine the type of line**: Since the inequality is \( \leq \), the line will be solid, indicating that points on the line are included in the solution set. 4. **Shade the appropriate region**: The inequality \( y \leq -\frac{3}{4} x - 1 \) indicates that we want the region below the line. To determine which side to shade, you can test a point not on the line, such as (0, 0): - Substitute (0, 0) into the inequality: \( 0 \leq -\frac{3}{4}(0) - 1 \) simplifies to \( 0 \leq -1 \), which is false. Therefore, we do not shade the region that includes (0, 0). 5. **Final graph**: The final graph will show a solid line representing \( y = -\frac{3}{4} x - 1 \) with the area below the line shaded, indicating all the points (x, y) that satisfy the inequality \( y \leq -\frac{3}{4} x - 1 \). This graph visually represents the solution set for the given inequality.