Certainly! Let's solve the equation [tex]\( |2x| = 14 \)[/tex] and illustrate the solution set on a number line.
Step 1: Understand the absolute value equation
The equation [tex]\( |2x| = 14 \)[/tex] tells us that the expression [tex]\( 2x \)[/tex] can be either positive or negative 14, because the absolute value of a number is the distance from zero and is always positive. Therefore, we split this equation into two separate cases.
Case 1: Positive scenario
[tex]\[ 2x = 14 \][/tex]
Step 2: Solve the equation for [tex]\( x \)[/tex]
[tex]\[ x = \frac{14}{2} \][/tex]
[tex]\[ x = 7 \][/tex]
Case 2: Negative scenario
[tex]\[ 2x = -14 \][/tex]
Step 3: Solve the equation for [tex]\( x \)[/tex]
[tex]\[ x = \frac{-14}{2} \][/tex]
[tex]\[ x = -7 \][/tex]
Step 4: Combine the solutions
The solutions from both cases are:
[tex]\[ x = 7 \][/tex]
[tex]\[ x = -7 \][/tex]
So, the solution set is:
[tex]\[ x = 7 \text{ and } x = -7 \][/tex]
Step 5: Represent the solution set on a number line
To represent this on a number line, we simply mark the points [tex]\( 7 \)[/tex] and [tex]\( -7 \)[/tex].
Here's a representation of the number line with these solutions:
```
<-------------------|-------------|-------------|-------------|-------------|------------------->
-10 -7 -5 0 5 7 10
```
The points [tex]\( -7 \)[/tex] and [tex]\( 7 \)[/tex] are the solutions to the equation [tex]\( |2x| = 14 \)[/tex].
