To solve the given absolute value equation [tex]\( |x - 2| = 6 \)[/tex], we need to find all possible values of [tex]\( x \)[/tex] that satisfy this equation. Here's a detailed step-by-step process to achieve this:
1. Understanding the Absolute Value Equation:
The absolute value [tex]\( |a| \)[/tex] can be defined as the distance of [tex]\( a \)[/tex] from 0 on the number line. If [tex]\( |x - 2| = 6 \)[/tex], this means that the distance between [tex]\( x \)[/tex] and 2 is 6 units. There are two scenarios to consider because absolute value equations split into two separate equations:
2. Setting Up the Two Scenarios:
- Scenario 1: [tex]\( x - 2 = 6 \)[/tex]
- Scenario 2: [tex]\( x - 2 = -6 \)[/tex]
3. Solving Each Scenario:
- For [tex]\( x - 2 = 6 \)[/tex]:
[tex]\[
x - 2 = 6
\][/tex]
Add 2 to both sides:
[tex]\[
x = 8
\][/tex]
- For [tex]\( x - 2 = -6 \)[/tex]:
[tex]\[
x - 2 = -6
\][/tex]
Add 2 to both sides:
[tex]\[
x = -4
\][/tex]
4. Solution Set:
The solutions to the equation [tex]\( |x-2|=6 \)[/tex] are [tex]\( x = 8 \)[/tex] and [tex]\( x = -4 \)[/tex].
Now, let's represent these solutions on a number line.
- On the number line, place a dot at [tex]\( x = 8 \)[/tex] because it's one of the solutions.
- Also, place a dot at [tex]\( x = -4 \)[/tex] because it is the other solution.
The number line will have two points indicated by dots at the positions [tex]\( x = 8 \)[/tex] and [tex]\( x = -4 \)[/tex].
Here's a rough sketch of the number line:
```
<---|----|----|----|----|----|----|---|---|---|---|---|---|---|--->
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
```
The asterisks (*) represent the solutions at [tex]\( x = -4 \)[/tex] and [tex]\( x = 8 \)[/tex].
This number line visually represents the solutions to the equation [tex]\( |x - 2| = 6 \)[/tex].
