To solve the system of inequalities by graphing, follow these step-by-step instructions:
1. Rewrite the Inequalities as Equations:
- Instead of inequalities, first, rewrite the given inequalities as equalities to find the boundary lines.
- [tex]\( y = -7x + 2 \)[/tex]
- [tex]\( y = 9x + 7 \)[/tex]
2. Plot the Boundary Lines:
- These equations represent the boundary lines of the regions described by the inequalities.
For [tex]\( y = -7x + 2 \)[/tex]:
- Determine two points on this line and connect them.
- When [tex]\( x = 0 \)[/tex], [tex]\( y = 2 \)[/tex]. One point is [tex]\((0, 2)\)[/tex].
- When [tex]\( x = 1 \)[/tex], [tex]\( y = -7(1) + 2 = -5 \)[/tex]. Another point is [tex]\((1, -5)\)[/tex].
For [tex]\( y = 9x + 7 \)[/tex]:
- Similarly, determine two points on this line and connect them.
- When [tex]\( x = 0 \)[/tex], [tex]\( y = 7 \)[/tex]. One point is [tex]\((0, 7)\)[/tex].
- When [tex]\( x = -1 \)[/tex], [tex]\( y = 9(-1) + 7 = -2 \)[/tex]. Another point is [tex]\((-1, -2)\)[/tex].
3. Shade the Appropriate Regions:
- The inequality [tex]\( y \leq -7x + 2 \)[/tex] indicates the region below or on the line [tex]\( y = -7x + 2 \)[/tex].
- The inequality [tex]\( y \geq 9x + 7 \)[/tex] indicates the region above or on the line [tex]\( y = 9x + 7 \)[/tex].
4. Identify the Intersection (Common Region):
- The solution to the system of inequalities is the region where the shaded areas overlap.
- Identify and shade the overlapping region that satisfies both inequalities.
5. Graphical Representation:
- Draw the boundary lines on a coordinate plane.
- Use a dashed line for each boundary if the sign doesn't include 'equal to' (strict inequalities); otherwise, use a solid line. Here, you will use solid lines because the inequalities include 'equal to' (≤ and ≥).
- Shade the region below the line [tex]\( y = -7x + 2 \)[/tex] and above the line [tex]\( y = 9x + 7 \)[/tex].
You can then visualize the graph with these insights:
- The line [tex]\( y = -7x + 2 \)[/tex] will slope downhill (negative slope) from the point [tex]\((0, 2)\)[/tex].
- The line [tex]\( y = 9x + 7 \)[/tex] will slope steeply uphill (positive slope) from the point [tex]\((0, 7)\)[/tex].
In the coordinate plane, the triangular region where both inequalities overlap is the solution region which lies below the line [tex]\( y = -7x + 2 \)[/tex] and above the line [tex]\( y = 9x + 7 \)[/tex].
By following these steps, you can accurately graph and identify the solution region for the system of inequalities given.
