To solve the inequality [tex]\( 18 < \frac{h - 14}{-1} \)[/tex] and graph the solution, we should follow these steps:
1. Simplify the Inequality:
The given inequality is:
[tex]\[
18 < \frac{h - 14}{-1}
\][/tex]
2. Deal with the Negative Sign:
Since the denominator is [tex]\(-1\)[/tex], this will flip the direction of the inequality when we multiply both sides by [tex]\(-1\)[/tex]:
[tex]\[
18 < \frac{h - 14}{-1} \implies 18 \cdot -1 > \frac{h - 14}{-1} \cdot -1
\][/tex]
This gives us:
[tex]\[
-18 > h - 14
\][/tex]
3. Isolate the Variable [tex]\( h \)[/tex]:
To isolate [tex]\( h \)[/tex], we add 14 to both sides of the inequality:
[tex]\[
-18 + 14 > h
\][/tex]
[tex]\[
-4 > h
\][/tex]
This can be rewritten as:
[tex]\[
h < -4
\][/tex]
4. Solution Interpretation:
The solution to the inequality [tex]\( 18 < \frac{h - 14}{-1} \)[/tex] is:
[tex]\[
h < -4
\][/tex]
5. Graph the Solution:
To graph the solution [tex]\( h < -4 \)[/tex]:
- Plot an open circle at [tex]\( h = -4 \)[/tex] on the number line to indicate that [tex]\(-4\)[/tex] is not included in the solution set.
- Shade the region to the left of [tex]\( h = -4 \)[/tex] to represent all values less than [tex]\(-4\)[/tex].
Here is a step-by-step sketching process:
- Draw a number line.
- Locate the point [tex]\(-4\)[/tex] on the number line.
- Place an open circle at [tex]\(-4\)[/tex] (the open circle indicates that [tex]\(-4\)[/tex] is not part of the solution).
- Shade the line to the left of [tex]\(-4\)[/tex] to indicate that all values less than [tex]\(-4\)[/tex] are part of the solution set.
The final graphical representation will show an open circle at [tex]\(-4\)[/tex] with the line shaded to the left, illustrating that [tex]\( h \)[/tex] can take any value less than [tex]\(-4\)[/tex].
