To solve the inequality [tex]\(-3x + y \leq 9\)[/tex], follow these steps:
1. Isolate [tex]\( y \)[/tex]:
To make it easier to understand the inequality, we can start by isolating [tex]\( y \)[/tex]. Simply add [tex]\( 3x \)[/tex] to both sides:
[tex]\[
y \leq 3x + 9
\][/tex]
2. Interpret the Inequality:
The inequality [tex]\( y \leq 3x + 9 \)[/tex] tells us that for any given value of [tex]\( x \)[/tex], [tex]\( y \)[/tex] should be less than or equal to [tex]\( 3x + 9 \)[/tex].
3. Graph the Boundary Line:
The boundary of the inequality is given by the line equation [tex]\( y = 3x + 9 \)[/tex]. To graph this line:
- Find the y-intercept by setting [tex]\( x = 0 \)[/tex]:
[tex]\[
y = 3(0) + 9 = 9
\][/tex]
So, the line crosses the y-axis at [tex]\( (0, 9) \)[/tex].
- Find another point by choosing another value for [tex]\( x \)[/tex]. For instance, if [tex]\( x = 1 \)[/tex]:
[tex]\[
y = 3(1) + 9 = 12
\][/tex]
So, another point on the line is [tex]\( (1, 12) \)[/tex].
Plot these points and draw the line [tex]\( y = 3x + 9 \)[/tex].
4. Determine the Shaded Region:
Since the inequality is [tex]\(\leq\)[/tex], we include the area below the line as well as the line itself. To determine this, pick a test point not on the line (the origin [tex]\( (0,0) \)[/tex] is usually a convenient choice):
- Substitute [tex]\( x = 0 \)[/tex] and [tex]\( y = 0 \)[/tex] into the inequality:
[tex]\[
0 \leq 3(0) + 9
\][/tex]
[tex]\[
0 \leq 9
\][/tex]
This statement is true, so the region that includes the origin is part of the solution region. Shade the area below (or on) the line [tex]\( y = 3x + 9 \)[/tex].
5. Summary:
The solution to the inequality [tex]\(-3x + y \leq 9\)[/tex] is the set of all points [tex]\((x, y)\)[/tex] such that [tex]\( y \)[/tex] is less than or equal to [tex]\( 3x + 9 \)[/tex]. Graphically, it is represented by the region below and including the line [tex]\( y = 3x + 9 \)[/tex].
