Sure, let's solve the inequality [tex]\(3x - 11 > 7x + 9\)[/tex] step-by-step and then represent the solution on a number line.
1. Combine Like Terms and Arrange the Inequality:
- We start with the inequality: [tex]\(3x - 11 > 7x + 9\)[/tex].
- Move all terms involving [tex]\(x\)[/tex] to one side of the inequality. Subtract [tex]\(7x\)[/tex] from both sides:
[tex]\[3x - 11 - 7x > 9\][/tex]
- Combine the [tex]\(x\)[/tex] terms:
[tex]\[-4x - 11 > 9\][/tex]
2. Isolate the Variable:
- Move all constant terms to the other side by adding 11 to both sides:
[tex]\[-4x - 11 + 11 > 9 + 11\][/tex]
- Combine the constants:
[tex]\[-4x > 20\][/tex]
3. Solve for [tex]\(x\)[/tex]:
- Divide both sides of the inequality by [tex]\(-4\)[/tex]. Note that dividing by a negative number reverses the inequality sign:
[tex]\[x < -5\][/tex]
So, the solution to the inequality [tex]\(3x - 11 > 7x + 9\)[/tex] is [tex]\(x < -5\)[/tex].
### Graphing the Solution on a Number Line
To graph [tex]\(x < -5\)[/tex] on a number line:
1. Draw a number line.
2. Mark the point [tex]\(-5\)[/tex] clearly on the number line.
3. Use an open circle (or parenthesis) on [tex]\(-5\)[/tex] to denote that [tex]\(-5\)[/tex] is not included in the solution set.
4. Shade or draw an arrow to the left of [tex]\(-5\)[/tex] to represent all numbers less than [tex]\(-5\)[/tex].
The number line should look something like this:
```
<---|---|---|---|---|---|---|---|---|---|--->
-8 -7 -6 (-5) -4 -3 -2 -1 0 1
<--- o====================>
```
Where [tex]\(( -5 )\)[/tex] represents the open circle at [tex]\(-5\)[/tex], indicating that [tex]\(-5\)[/tex] is not included in the solution, and the arrow to the left indicates all numbers less than [tex]\(-5\)[/tex].
