Certainly! Let's solve the inequality step-by-step first, and then graph the solution on the number line.
### Inequality:
[tex]\[ 3x - 11 > 7x + 9 \][/tex]
### Step 1: Move all terms involving [tex]\(x\)[/tex] to one side of the inequality.
Subtract [tex]\(7x\)[/tex] from both sides:
[tex]\[ 3x - 11 - 7x > 7x + 9 - 7x \][/tex]
This simplifies to:
[tex]\[ -4x - 11 > 9 \][/tex]
### Step 2: Move the constant term to the other side of the inequality.
Add 11 to both sides:
[tex]\[ -4x - 11 + 11 > 9 + 11 \][/tex]
This simplifies to:
[tex]\[ -4x > 20 \][/tex]
### Step 3: Solve for [tex]\(x\)[/tex].
Divide both sides by -4. Remember, dividing an inequality by a negative number reverses the inequality sign:
[tex]\[ x < \frac{20}{-4} \][/tex]
[tex]\[ x < -5 \][/tex]
### Solution:
The solution to the inequality [tex]\(3x - 11 > 7x + 9\)[/tex] is [tex]\(x < -5\)[/tex].
### Graphing the solution on a number line:
1. We need to indicate all numbers less than [tex]\(-5\)[/tex].
2. Use an open point (circle) to represent [tex]\(-5\)[/tex], since [tex]\(-5\)[/tex] is not included in the solution.
3. Draw a ray extending to the left from [tex]\(-5\)[/tex], indicating all numbers less than [tex]\(-5\)[/tex].
Here is How you would graph it:
```
(-∞)------------------(-5)==============================(∞)
∘
```
- The open circle (∘) at -5 indicates that -5 is not included.
- The arrow extending to the left shows that the solution includes all numbers less than -5.
This completes the graphing of the solution set on the number line.
