Certainly! Let's consider the inequality [tex]\( y \geq \frac{9}{4} x - 5 \)[/tex]. This inequality describes a region in the coordinate plane.
### Step-by-Step Solution:
1. Understanding the Inequality:
- The inequality [tex]\( y \geq \frac{9}{4} x - 5 \)[/tex] means that for any point [tex]\((x, y)\)[/tex] in the coordinate plane, the y-coordinate of the point should be greater than or equal to the expression [tex]\( \frac{9}{4} x - 5 \)[/tex].
2. Graphing the Boundary Line:
- First, let's graph the boundary line represented by the equation [tex]\( y = \frac{9}{4} x - 5 \)[/tex]. This is a straight line with a slope of [tex]\(\frac{9}{4}\)[/tex] and a y-intercept of [tex]\(-5\)[/tex].
- Slope: [tex]\(\frac{9}{4}\)[/tex] tells us that for every increase of 4 units in the x-direction, [tex]\( y \)[/tex] increases by 9 units.
- Y-intercept: The line crosses the y-axis at [tex]\(-5\)[/tex] (i.e., the point [tex]\((0, -5)\)[/tex]).
3. Shading the Region:
- The inequality symbol [tex]\( \geq \)[/tex] indicates that the region of interest includes the line itself (since [tex]\( y \)[/tex] can be equal to [tex]\( \frac{9}{4} x - 5 \)[/tex]), as well as the region above this line.
- To shade the correct region, choose a test point that is not on the boundary line, for instance, [tex]\((0, 0)\)[/tex].
- Substitute [tex]\((0, 0)\)[/tex] into the inequality: [tex]\( 0 \geq \frac{9}{4} \cdot 0 - 5 \)[/tex]
- Simplifies to: [tex]\( 0 \geq -5 \)[/tex], which is true.
- Hence, [tex]\((0, 0)\)[/tex] is in the region satisfying the inequality, so we shade the area above and including the line [tex]\( y = \frac{9}{4} x - 5 \)[/tex].
4. Verify a Specific Point:
- Let's verify if a specific point [tex]\((4, 6)\)[/tex] lies within the region defined by the inequality:
- Substitute [tex]\( x = 4\)[/tex] and [tex]\( y = 6\)[/tex] into the inequality:
[tex]\[
6 \geq \frac{9}{4} \cdot 4 - 5
\][/tex]
- Simplify the right side:
[tex]\[
6 \geq 9 - 5
\][/tex]
[tex]\[
6 \geq 4
\][/tex]
- This is true.
- Therefore, the point [tex]\((4, 6)\)[/tex] lies within the region defined by the inequality [tex]\( y \geq \frac{9}{4} x - 5 \)[/tex].
By following these steps, you can determine that the inequality [tex]\( y \geq \frac{9}{4} x - 5 \)[/tex] defines a region on the coordinate plane including the boundary line and the area above it.
