Sure, let's go through the steps to graph the inequality [tex]\( y \leq -2x - 1 \)[/tex].
### Step-by-Step Solution
1. Understand the inequality:
- The inequality is given as [tex]\( y \leq -2x - 1 \)[/tex].
- This means we are looking for all the points [tex]\((x, y)\)[/tex] in the plane where the y-coordinate is less than or equal to [tex]\( -2x - 1 \)[/tex].
2. Graph the boundary line:
- First, we graph the boundary line [tex]\( y = -2x - 1 \)[/tex]. This line runs through the plane, and we will use it to help us graph the inequality.
- To graph this, we need at least two points.
3. Find points for the boundary line:
- When [tex]\( x = 0 \)[/tex], [tex]\( y = -2(0) - 1 = -1 \)[/tex]. So, one point is [tex]\((0, -1)\)[/tex].
- When [tex]\( x = 1 \)[/tex], [tex]\( y = -2(1) - 1 = -3 \)[/tex]. So, another point is [tex]\((1, -3)\)[/tex].
4. Plot the boundary line:
- Plot the points [tex]\((0, -1)\)[/tex] and [tex]\((1, -3)\)[/tex] on the graph.
- Draw a straight line through these points. This line represents the equation [tex]\( y = -2x - 1 \)[/tex].
5. Determine the region to shade:
- The inequality is [tex]\( y \leq -2x - 1 \)[/tex]. This means we want the region where y is less than or equal to the boundary line.
- To determine the correct region to shade, choose a test point that is not on the boundary line (the origin [tex]\((0,0)\)[/tex] is often convenient unless it lies on the boundary).
6. Test the point (0,0):
- Substitute [tex]\((0,0)\)[/tex] into the inequality: [tex]\( 0 \leq -2(0) - 1 \)[/tex].
- Simplifies to [tex]\( 0 \leq -1 \)[/tex], which is false.
- Since [tex]\((0,0)\)[/tex] does not satisfy the inequality, the region that does will be on the other side of the line.
7. Shade the correct region:
- Shade the region below the line [tex]\( y = -2x - 1 \)[/tex], as that represents [tex]\( y \leq -2x - 1 \)[/tex].
### Final Graph
- The boundary line [tex]\( y = -2x - 1 \)[/tex] is a straight line with a negative slope.
- The area below this line, including the line itself (since it's [tex]\( \leq \)[/tex]), is shaded.
The graphical representation would look like this:
- A straight line passing through [tex]\((0, -1)\)[/tex] and [tex]\((1, -3)\)[/tex].
- The area below this line shaded to represent all points [tex]\((x, y)\)[/tex] where [tex]\( y \leq -2x - 1 \)[/tex].
### Conclusion
The correct graph for the inequality [tex]\( y \leq -2x - 1 \)[/tex] is a line with a slope of -2, passing through the point [tex]\((0, -1)\)[/tex] and the region below the line shaded.
