To solve the equation [tex]\(||x - 1| - 2| = |x - 3|\)[/tex], we need to consider the behaviors of the absolute value functions within the equation. The absolute value function splits into different cases based on the ranges of input values, so we need to evaluate all possible cases.
### Step-by-Step Solution
1. Break down the absolute values:
The equation is [tex]\(||x - 1| - 2| = |x - 3|\)[/tex]. We start by analyzing the inner absolute value function [tex]\(|x - 1|\)[/tex]:
- Case 1: [tex]\(x \geq 1\)[/tex]
[tex]\[
|x - 1| = x - 1
\][/tex]
- Case 2: [tex]\(x < 1\)[/tex]
[tex]\[
|x - 1| = 1 - x
\][/tex]
2. Substitute the cases into the equation [tex]\(||x - 1| - 2| = |x - 3|\)[/tex]:
- For [tex]\(x \geq 1\)[/tex]:
[tex]\[
|x - 1| = x - 1
\][/tex]
Substituting this into the original equation, we get:
[tex]\[
||x - 1| - 2| = |x - 3|
\][/tex]
Which simplifies to:
[tex]\[
|(x - 1) - 2| = |x - 3|
\][/tex]
Thus:
[tex]\[
|x - 3| = |x - 3|
\][/tex]
This is true for all [tex]\(x \geq 1\)[/tex].
- For [tex]\(x < 1\)[/tex]:
[tex]\[
|x - 1| = 1 - x
\][/tex]
Substituting this into the original equation, we get:
[tex]\[
||1 - x| - 2| = |x - 3|
\][/tex]
Which simplifies to:
[tex]\[
|(1 - x) - 2| = |x - 3|
\][/tex]
Thus:
[tex]\[
|1 - x - 2| = |x - 3|
\][/tex]
Simplifying inside the absolute value:
[tex]\[
|1 - x - 2| = |x - 3|
\][/tex]
Simplifies to:
[tex]\[
|-(x + 1)| = |x - 3|
\][/tex]
Simplifies further to:
[tex]\[
|x + 1| = |x - 3|
\][/tex]
3. Break the resulting absolute values into additional cases:
[tex]\[
|x + 1| = |x - 3|
\][/tex]
- Case A: [tex]\(x + 1 = x - 3\)[/tex]
[tex]\[
1 = -3
\][/tex]
This is a contradiction and has no solution.
- Case B: [tex]\(x + 1 = -(x - 3)\)[/tex]
[tex]\[
x + 1 = -x + 3
\][/tex]
Solving for [tex]\(x\)[/tex]:
[tex]\[
x + x = 3 - 1
\][/tex]
[tex]\[
2x = 2
\][/tex]
[tex]\[
x = 1
\][/tex]
- Case C: [tex]\(x \geq 1\)[/tex]
As previously established, [tex]\(|x - 3| = |x - 3|\)[/tex] is always true.
Combining all these results, we see that when verifying back into the original equation, [tex]\(x = 1\)[/tex] was already covered under the condition [tex]\(x \geq 1\)[/tex].
Therefore, the solutions to the given equation are:
### Solution:
[tex]\[
x \geq 1
\][/tex]
To summarize, the step-by-step approach yielded that valid solution based on analyzing cases of absolute values. Solving [tex]\(||x - 1| - 2| = |x - 3|\)[/tex] confirms that [tex]\(x \geq 1\)[/tex].
