To solve the system of inequalities:
[tex]\[
\left\{
\begin{array}{l}
y \leq -3x - 2 \\
y > x - 2
\end{array}
\right.
\][/tex]
we can follow these steps:
1. Identify the boundary lines for each inequality:
For the first inequality [tex]\( y \leq -3x - 2 \)[/tex], the boundary line is given by:
[tex]\[
y = -3x - 2
\][/tex]
This line has a slope of [tex]\(-3\)[/tex] and a y-intercept at [tex]\(-2\)[/tex].
For the second inequality [tex]\( y > x - 2 \)[/tex], the boundary line is given by:
[tex]\[
y = x - 2
\][/tex]
This line has a slope of [tex]\(1\)[/tex] and a y-intercept at [tex]\(-2\)[/tex].
2. Graph the boundary lines:
- The line [tex]\( y = -3x - 2 \)[/tex] slopes downward from left to right. Start at the point [tex]\((0, -2)\)[/tex], move down 3 units and right 1 unit to find another point, and draw the line through these points.
- The line [tex]\( y = x - 2 \)[/tex] slopes upward from left to right. Start at the point [tex]\((0, -2)\)[/tex], move up 1 unit and right 1 unit to find another point, and draw the line through these points.
3. Determine the regions for the inequalities:
- For [tex]\( y \leq -3x - 2 \)[/tex]: The region below and including the line [tex]\( y = -3x - 2 \)[/tex].
- For [tex]\( y > x - 2 \)[/tex]: The region above the line [tex]\( y = x - 2 \)[/tex].
4. Find the region that satisfies both inequalities:
Combine the regions determined from the two inequalities:
- The solution region is where the shaded areas of both inequalities overlap.
Let's visualize this on a graph:
- Sketch the line [tex]\( y = -3x - 2 \)[/tex]. Shade the region below this line (up to and including the line).
- Sketch the line [tex]\( y = x - 2 \)[/tex]. Shade the region above this line.
The intersection of these shaded regions will form a wedge-shaped area. This region represents all the points [tex]\((x, y)\)[/tex] that satisfy both inequalities.
Therefore, the solution to the system of inequalities [tex]\(\left\{\begin{array}{l}y \leq -3x - 2 \\ y > x - 2\end{array}\right.\)[/tex] is the area where the two shaded regions overlap. This region can be graphically represented and is defined by the boundary lines:
[tex]\[
y = -3x - 2 \quad \text{and} \quad y = x - 2
\][/tex]
These boundaries help us clearly understand the limits within which the solution lies.
