Certainly! Let's solve the inequality step-by-step to understand what it represents.
1. Understanding the Inequality:
The given inequality is [tex]\( y \geq -\frac{1}{2}x + 1 \)[/tex]. This represents a region in the Cartesian plane.
2. Slope-Intercept Form:
Our inequality is already in slope-intercept form, [tex]\( y \geq mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
- The slope ([tex]\( m \)[/tex]) is [tex]\(-\frac{1}{2}\)[/tex].
- The y-intercept ([tex]\( b \)[/tex]) is [tex]\( 1 \)[/tex].
3. Graphing the Boundary Line:
- Y-Intercept: Start by plotting the y-intercept (0, 1) on the graph.
- Using the Slope: From the y-intercept (0, 1), use the slope [tex]\(-\frac{1}{2}\)[/tex] to find another point. The slope [tex]\(-\frac{1}{2}\)[/tex] means for every 2 units you move to the right (positive x direction), you move 1 unit down (negative y direction).
So, starting from (0, 1):
- Move 2 units to the right (x = 2).
- Move 1 unit down (y = 0).
This gives you another point at (2, 0).
- Draw the line through these points: Connect (0, 1) and (2, 0) with a straight line. This line represents [tex]\( y = -\frac{1}{2}x + 1 \)[/tex].
4. Shading the Region:
The inequality [tex]\( y \geq -\frac{1}{2}x + 1 \)[/tex] means we need to shade the region that is above the line or on the line.
- To determine which side to shade, pick a test point that is not on the line (the origin (0, 0) is often a convenient choice):
Substitute [tex]\( x = 0 \)[/tex] and [tex]\( y = 0 \)[/tex] into the inequality [tex]\( y \geq -\frac{1}{2}x + 1 \)[/tex]:
[tex]\[
0 \geq -\frac{1}{2}(0) + 1
\][/tex]
[tex]\[
0 \geq 1
\][/tex]
This statement is false, so the origin (0, 0) is not in the solution set. This indicates the region that does not include the origin is the solution set.
- Therefore, shade the region above the line [tex]\( y = -\frac{1}{2}x + 1 \)[/tex].
5. In Conclusion:
The inequality [tex]\( y \geq -\frac{1}{2}x + 1 \)[/tex] signifies that for any point [tex]\( (x, y) \)[/tex] in the solution set, the y-value must be greater than or equal to [tex]\( -\frac{1}{2}x + 1 \)[/tex].
Thus, we have:
[tex]\[
\boxed{y \geq -\frac{1}{2}x + 1}
\][/tex]
