To solve the inequality [tex]\( 3x - 11 > 7x + 9 \)[/tex], follow these steps:
1. Isolate the variable [tex]\( x \)[/tex] on one side:
[tex]\[
3x - 11 > 7x + 9
\][/tex]
2. Subtract [tex]\( 7x \)[/tex] from both sides to get:
[tex]\[
3x - 7x - 11 > 9
\][/tex]
This simplifies to:
[tex]\[
-4x - 11 > 9
\][/tex]
3. Add 11 to both sides to get:
[tex]\[
-4x - 11 + 11 > 9 + 11
\][/tex]
This simplifies to:
[tex]\[
-4x > 20
\][/tex]
4. Divide both sides by -4 (and remember to flip the inequality sign because you are dividing by a negative number):
[tex]\[
x < \frac{20}{-4}
\][/tex]
This simplifies to:
[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 represent [tex]\( x < -5 \)[/tex]:
1. Draw a number line.
2. Identify the point -5 on the number line.
3. Draw an open circle (or dot) at -5, because -5 is not included in the solution set.
4. Shade or draw a line to the left of -5, indicating all values less than -5.
Here's a visual representation on a number line:
```
--------|--------|--------|--------|--------|--------|--------|--------|--------
-7 -6 -5 -4 -3 -2 -1 0
<-----------------------
```
Notice that the solution includes all values to the left of -5 on the number line, but not -5 itself, as indicated by the open circle.
Thus, the solution set [tex]\( x < -5 \)[/tex] is correctly graphed on the number line.
