To solve the inequality
[tex]\[
-6(4x - 2) > -108
\][/tex]
we need to follow a series of algebraic steps to simplify and solve for [tex]\( x \)[/tex].
1. Distribute the -6:
We start by distributing the -6 across the terms inside the parentheses:
[tex]\[
-6(4x - 2) = -24x + 12
\][/tex]
So the inequality now becomes:
[tex]\[
-24x + 12 > -108
\][/tex]
2. Isolate the [tex]\( x \)[/tex]-term:
Next, we need to isolate the term involving [tex]\( x \)[/tex]. Begin by subtracting 12 from both sides of the inequality:
[tex]\[
-24x + 12 - 12 > -108 - 12
\][/tex]
Simplifying this, we get:
[tex]\[
-24x > -120
\][/tex]
3. Solve for [tex]\( x \)[/tex]:
Now, we need to solve for [tex]\( x \)[/tex] by dividing both sides of the inequality by -24. Remember, when you divide or multiply both sides of an inequality by a negative number, the inequality sign flips:
[tex]\[
x < \frac{-120}{-24}
\][/tex]
Simplifying this, we get:
[tex]\[
x < 5
\][/tex]
4. Write the solution in interval notation:
The solution in interval notation for the inequality [tex]\( x < 5 \)[/tex] is:
[tex]\[
(-\infty, 5)
\][/tex]
Therefore, the solution to the inequality is [tex]\( x < 5 \)[/tex].
To graph this solution on a number line:
- We draw a number line with appropriate divisions.
- Mark the point 5 on the line with an open circle to indicate that 5 is not included in the solution.
- Shade the region to the left of 5 to show all the values that are less than 5.
The graphical representation is as follows:
```
<--------|------>
5
```
This shaded region indicates that [tex]\( x \)[/tex] can be any value less than 5.
