To solve the inequality [tex]\( x + 2y > 9 \)[/tex], let's go through a detailed, step-by-step explanation of what this inequality represents and how to interpret it:
1. Identify the Boundary Line:
- Start by considering the related equation [tex]\( x + 2y = 9 \)[/tex]. This is the boundary line that separates the region where the inequality is either true or false.
2. Graph the Boundary Line:
- To graph [tex]\( x + 2y = 9 \)[/tex], find two points that lie on this line:
- When [tex]\( y = 0 \)[/tex]:
[tex]\[
x + 2(0) = 9 \implies x = 9
\][/tex]
So, one point is [tex]\( (9, 0) \)[/tex].
- When [tex]\( x = 0 \)[/tex]:
[tex]\[
0 + 2y = 9 \implies y = \frac{9}{2} = 4.5
\][/tex]
So, another point is [tex]\( (0, 4.5) \)[/tex].
- Plot these points on a coordinate plane and draw a straight line through them. This line represents the equation [tex]\( x + 2y = 9 \)[/tex].
3. Determine the Half-Plane:
- The inequality [tex]\( x + 2y > 9 \)[/tex] indicates that we are interested in the region above the line [tex]\( x + 2y = 9 \)[/tex]. To determine which side of the line this is, we can test a point that is not on the line (a simple one is the origin [tex]\( (0,0) \)[/tex]):
[tex]\[
0 + 2(0) \leq 9
\][/tex]
Since [tex]\( 0 \leq 9 \)[/tex] is false, the region that satisfies [tex]\( x + 2y > 9 \)[/tex] is not on the side of the origin; it is on the opposite side of the line.
4. Shade the Region:
- The inequality [tex]\( x + 2y > 9 \)[/tex] represents all the points above the line [tex]\( x + 2y = 9 \)[/tex]. Draw a dotted line for [tex]\( x + 2y = 9 \)[/tex] and shade the region above this line. Remember that the line itself is not included in the solution (represented by using a dashed line).
In summary, the inequality [tex]\( x + 2y > 9 \)[/tex] describes a half-plane in the coordinate system that lies above the line [tex]\( x + 2y = 9 \)[/tex]. This region consists of all points [tex]\((x, y)\)[/tex] that satisfy the condition [tex]\( x + 2y > 9 \)[/tex]. The boundary line [tex]\( x + 2y = 9 \)[/tex] itself is not included in the solution.
