Let's solve the equation [tex]\( |2x| = 14 \)[/tex] step-by-step.
### Step 1: Understand the Absolute Value Equation
The absolute value equation [tex]\( |2x| = 14 \)[/tex] means that the expression inside the absolute value, [tex]\( 2x \)[/tex], can be either positive or negative. This gives us two separate equations to consider:
1. [tex]\( 2x = 14 \)[/tex]
2. [tex]\( 2x = -14 \)[/tex]
### Step 2: Solve Each Equation
#### Equation 1: [tex]\( 2x = 14 \)[/tex]
Divide both sides by 2 to solve for [tex]\( x \)[/tex]:
[tex]\[
2x = 14 \implies x = \frac{14}{2} = 7
\][/tex]
#### Equation 2: [tex]\( 2x = -14 \)[/tex]
Again, divide both sides by 2 to solve for [tex]\( x \)[/tex]:
[tex]\[
2x = -14 \implies x = \frac{-14}{2} = -7
\][/tex]
### Step 3: Combine the Solutions
The solutions from both equations are:
[tex]\[
x = 7 \quad \text{and} \quad x = -7
\][/tex]
Thus, the solution set is:
[tex]\[
\{7, -7\}
\][/tex]
### Step 4: Show the Solution Set on a Number Line
Draw a number line and plot the points [tex]\(7\)[/tex] and [tex]\(-7\)[/tex]:
```
----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|----|
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
-7 7
```
### Step 5: Check the Solutions
Substitute the solutions back into the original absolute value equation to verify.
For [tex]\( x = 7 \)[/tex]:
[tex]\[
|2 \cdot 7| = |14| = 14 \quad \text{(True)}
\][/tex]
For [tex]\( x = -7 \)[/tex]:
[tex]\[
|2 \cdot (-7)| = |-14| = 14 \quad \text{(True)}
\][/tex]
Both solutions are correct. Therefore, the final solution set is [tex]\( \{7, -7\} \)[/tex].
