Let's solve the linear inequality [tex]\( x - 2y \geq -12 \)[/tex] and understand how to graph it step by step.
1. Isolate [tex]\( y \)[/tex] in the inequality:
[tex]\[
x - 2y \geq -12
\][/tex]
Subtract [tex]\( x \)[/tex] from both sides:
[tex]\[
-2y \geq -x - 12
\][/tex]
Divide each term by -2. Remember to reverse the inequality sign when dividing by a negative number:
[tex]\[
y \leq \frac{x + 12}{2}
\][/tex]
2. Identify the boundary line:
The corresponding equation of the boundary line is:
[tex]\[
y = \frac{x + 12}{2}
\][/tex]
This boundary line has a slope and a y-intercept. The slope of the line is the coefficient of [tex]\( x \)[/tex], which is [tex]\( \frac{1}{2} \)[/tex]. The y-intercept is the constant term, which is [tex]\( \frac{12}{2} = 6 \)[/tex].
3. Plot the boundary line:
- The y-intercept (where [tex]\( x = 0 \)[/tex]) is at [tex]\( (0, 6) \)[/tex].
- Use the slope to find another point. Slope [tex]\( \frac{1}{2} \)[/tex] means that for each increase of 2 units in [tex]\( x \)[/tex], [tex]\( y \)[/tex] increases by 1 unit. So, another point is at [tex]\( (2, 7) \)[/tex].
4. Draw the boundary line:
- Draw a straight line through the points [tex]\( (0, 6) \)[/tex] and [tex]\( (2, 7) \)[/tex]. This line represents [tex]\( y = \frac{x + 12}{2} \)[/tex].
5. Determine which side to shade:
- Since the inequality is [tex]\( y \leq \frac{x + 12}{2} \)[/tex], we need to shade the region below or on the line.
To confirm this:
- Choose a test point not on the line, such as the origin [tex]\( (0, 0) \)[/tex].
- Substitute [tex]\( (0, 0) \)[/tex] into [tex]\( y \leq \frac{x + 12}{2} \)[/tex]:
[tex]\[
0 \leq \frac{0 + 12}{2} \Rightarrow 0 \leq 6
\][/tex]
- The inequality holds true, so the region that includes [tex]\( (0,0) \)[/tex] (below the line) is the solution region.
6. Graph the solution:
- Draw the boundary line [tex]\( y = \frac{x + 12}{2} \)[/tex], which passes through [tex]\( (0, 6) \)[/tex] and [tex]\( (2, 7) \)[/tex], as a solid line because the inequality is "greater than or equal to."
- Shade the region below the line to represent all points [tex]\( (x, y) \)[/tex] that satisfy [tex]\( y \leq \frac{x + 12}{2} \)[/tex] or [tex]\( x - 2y \geq -12 \)[/tex].
The final graph shows the shaded region below the line [tex]\( y = \frac{x + 12}{2} \)[/tex], including the line itself, as the solution set for the inequality [tex]\( x - 2y \geq -12 \)[/tex].
