To determine the solution to the system of inequalities [tex]\( y \geq 2x - 3 \)[/tex] and [tex]\( y < 2x + 4 \)[/tex], we need to analyze the region that satisfies both conditions.
### Step-by-Step Solution
1. Understand the Inequalities:
- [tex]\( y \geq 2x - 3 \)[/tex]: This inequality represents the region that is on or above the line [tex]\( y = 2x - 3 \)[/tex].
- [tex]\( y < 2x + 4 \)[/tex]: This inequality represents the region that is below the line [tex]\( y = 2x + 4 \)[/tex].
2. Identify the Lines:
- The first line is [tex]\( y = 2x - 3 \)[/tex].
- The second line is [tex]\( y = 2x + 4 \)[/tex].
3. Plotting the Lines:
- We'll plot both lines on a coordinate plane.
- The line [tex]\( y = 2x - 3 \)[/tex] will be a solid line because the inequality [tex]\( y \geq 2x - 3 \)[/tex] includes the boundary.
- The line [tex]\( y = 2x + 4 \)[/tex] will be a dashed line because the inequality [tex]\( y < 2x + 4 \)[/tex] does not include the boundary.
4. Determine the Region of Interest:
- The region of interest is the set of points that lie above or on the line [tex]\( y = 2x - 3 \)[/tex] and below the line [tex]\( y = 2x + 4 \)[/tex].
- This region is bounded between the two lines.
5. Test Points to Verify the Region:
- For instance, if we choose a point between the lines, such as [tex]\((0, 0)\)[/tex]:
- Check [tex]\( y = 0 \geq 2(0) - 3 \implies 0 \geq -3 \)[/tex] (True)
- Check [tex]\( y = 0 < 2(0) + 4 \implies 0 < 4 \)[/tex] (True)
- This point satisfies both inequalities, confirming it lies in the solution region.
6. Conclusion:
- The combined region consists of all points between [tex]\( y = 2x - 3 \)[/tex] (inclusive) and [tex]\( y = 2x + 4 \)[/tex] (exclusive) as we vary [tex]\( x \)[/tex].
By examining the lines and the inequalities, we can visualize the solution graphically. This region should look like a band or strip between two lines flowing upward diagonally, with one boundary being a solid line and the other a dashed line.
