To graph the inequality [tex]\( y \geq 5x + 10 \)[/tex], follow these steps:
1. Graph the boundary line [tex]\( y = 5x + 10 \)[/tex]:
- This line is the equality part of the inequality.
- For this line, we need at least two points to draw it accurately.
2. Find the y-intercept:
- When [tex]\( x = 0 \)[/tex], [tex]\( y = 5(0) + 10 = 10 \)[/tex].
- This gives us the point (0, 10), which is the y-intercept.
3. Find another point using the slope:
- The slope of the line [tex]\( y = 5x + 10 \)[/tex] is 5, meaning for each increase of 1 unit in [tex]\( x \)[/tex], [tex]\( y \)[/tex] increases by 5 units.
- Starting at the y-intercept (0, 10), if [tex]\( x = 1 \)[/tex], then [tex]\( y = 5(1) + 10 = 15 \)[/tex].
- This gives us the point (1, 15).
4. Plot these points on the coordinate plane:
- Plot the points (0, 10) and (1, 15).
5. Draw the boundary line:
- Connect these two points with a straight line.
- Since the inequality is [tex]\( y \geq 5x + 10 \)[/tex] and not just [tex]\( y = 5x + 10 \)[/tex], the line should be solid, indicating that points on the line satisfy the inequality.
6. Shade the region representing the inequality [tex]\( y \geq 5x + 10 \)[/tex]:
- Identify which side of the line to shade by picking a test point that is not on the boundary line. A convenient test point is the origin (0, 0).
- Substitute (0, 0) into the inequality to see if it holds true:
[tex]\[
0 \geq 5(0) + 10
\Rightarrow 0 \geq 10
\][/tex]
- This is false, so the region that does not include (0, 0) should be shaded.
7. Shade above the line:
- The correct region to shade is above the line [tex]\( y = 5x + 10 \)[/tex]. This represents all points [tex]\((x, y)\)[/tex] where [tex]\( y \)[/tex] is greater than or equal to [tex]\( 5x + 10 \)[/tex].
The final graph will have a solid line passing through (0, 10) and (1, 15) with the area above this line shaded. This visualizes all points that satisfy the inequality [tex]\( y \geq 5x + 10 \)[/tex].
