To graph the solution to the given system of inequalities:
[tex]\[ \begin{array}{l}
y \leq -3x - 4 \\
y > 3x + 7
\end{array} \][/tex]
follow these steps:
### Step 1: Graph the boundary lines
1. Draw the line for [tex]\( y = -3x - 4 \)[/tex]:
- This line is in slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m = -3 \)[/tex] (slope) and [tex]\( b = -4 \)[/tex] (y-intercept).
- Plot the y-intercept [tex]\((0, -4)\)[/tex].
- Use the slope to find another point. For example, starting at [tex]\((0, -4)\)[/tex], move down 3 units and right 1 unit to get to the point [tex]\((1, -7)\)[/tex].
- Draw a solid line through these points because the inequality includes [tex]\( \leq \)[/tex] (less than or equal to), which means points on the line are part of the solution.
2. Draw the line for [tex]\( y = 3x + 7 \)[/tex]:
- This line is also in slope-intercept form with [tex]\( m = 3 \)[/tex] (slope) and [tex]\( b = 7 \)[/tex] (y-intercept).
- Plot the y-intercept [tex]\((0, 7)\)[/tex].
- Use the slope to find another point. For example, starting at [tex]\((0, 7)\)[/tex], move up 3 units and right 1 unit to get to the point [tex]\((1, 10)\)[/tex].
- Draw a dashed line through these points because the inequality includes [tex]\( > \)[/tex] (greater than), which means points on the line are not part of the solution.
### Step 2: Shade the appropriate regions
1. Shade the region for [tex]\( y \leq -3x - 4 \)[/tex]:
- This means you need to shade below the line [tex]\( y = -3x - 4 \)[/tex]. Any point below or on this line satisfies the inequality.
2. Shade the region for [tex]\( y > 3x + 7 \)[/tex]:
- This means you need to shade above the line [tex]\( y = 3x + 7 \)[/tex]. Any point above this line satisfies the inequality.
### Step 3: Identify the solution region
- The solution to the system of inequalities is the region where the shaded areas from the two inequalities do not overlap.
- In this case, the lines [tex]\( y = -3x - 4 \)[/tex] and [tex]\( y = 3x + 7 \)[/tex] diverge and their respective shaded regions do not intersect. Thus, there are no common points that satisfy both inequalities.
### Graph Summary
- Draw a solid line representing [tex]\( y = -3x - 4 \)[/tex].
- Draw a dashed line representing [tex]\( y = 3x + 7 \)[/tex].
- Shade below the solid line [tex]\( y = -3x - 4 \)[/tex].
- Shade above the dashed line [tex]\( y = 3x + 7 \)[/tex].
- Note that there is no overlap between these shaded regions, indicating there is no set of points that satisfies both inequalities simultaneously.
Ultimately, the system of inequalities has no solution because there is no overlapping region that satisfies both conditions.
