To solve the inequality [tex]\(3x - 11 > 7x + 9\)[/tex], let's work through it step-by-step.
1. First, isolate the variable [tex]\(x\)[/tex]:
[tex]\[
3x - 11 > 7x + 9
\][/tex]
2. Subtract [tex]\(7x\)[/tex] from both sides to start moving the terms involving [tex]\(x\)[/tex] to one side:
[tex]\[
3x - 7x - 11 > 9
\][/tex]
3. Combine like terms on the left side:
[tex]\[
-4x - 11 > 9
\][/tex]
4. Add 11 to both sides to move the constant term to the right side:
[tex]\[
-4x - 11 + 11 > 9 + 11
\][/tex]
[tex]\[
-4x > 20
\][/tex]
5. Divide both sides by -4, remembering that dividing by a negative number reverses the inequality:
[tex]\[
x < \frac{20}{-4}
\][/tex]
[tex]\[
x < -5
\][/tex]
So the solution is [tex]\(x < -5\)[/tex].
### Graphing the Solution on a Number Line
To graph the solution [tex]\(x < -5\)[/tex] on a number line:
1. Draw a number line.
2. Locate and mark the point [tex]\(-5\)[/tex] on the number line.
3. Draw an open circle at [tex]\(-5\)[/tex] to indicate that [tex]\(-5\)[/tex] is not included in the solution set.
4. Shade the number line to the left of [tex]\(-5\)[/tex] to represent all values less than [tex]\(-5\)[/tex].
Here’s what it looks like:
```
<-------(------->
-5
```
- The open circle at [tex]\(-5\)[/tex] shows that [tex]\(-5\)[/tex] is not included.
- The line going to the left from [tex]\(-5\)[/tex] represents all numbers less than [tex]\(-5\)[/tex].
So, the solution set is graphically represented by shading to the left of [tex]\(-5\)[/tex] with an open circle at [tex]\(-5\)[/tex].
