Question

Solve these equations. Show the solutions on a number line.
|18-x| = 3

Answer 1

To solve the equation \(|18 - x| = 3\), we will consider the definition of absolute value. The equation can be broken down into two separate cases:

1. **Case 1**: \(18 - x = 3\)
2. **Case 2**: \(18 - x = -3\)

Let's solve each case.

### Case 1: \(18 - x = 3\)

To solve for \(x\):
\[
18 - x = 3
\]
Subtract 18 from both sides:
\[
-x = 3 - 18
\]
\[
-x = -15
\]
Multiply both sides by -1:
\[
x = 15
\]

### Case 2: \(18 - x = -3\)

Now, solving this case:
\[
18 - x = -3
\]
Subtract 18 from both sides:
\[
-x = -3 - 18
\]
\[
-x = -21
\]
Multiply both sides by -1:
\[
x = 21
\]

### Solutions
The solutions to the original equation \(|18 - x| = 3\) are:
\[
x = 15 \quad \text{and} \quad x = 21
\]

### Number Line Representation
To represent these solutions on a number line, we can plot points at \(x = 15\) and \(x = 21\):

```
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
0 5 10 15 20 25 30
●
●
```
In this visualization, we have:
- A dot at \(15\) (●) indicating one solution.
- A dot at \(21\) (●) indicating the other solution.

These marks show where the solutions to the equation are located on the number line.