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What is the solution of [tex]|x-4|-10=1[/tex]?

A. [tex]x=-15[/tex] or [tex]x=15[/tex]

B. [tex]x=-7[/tex] or [tex]x=15[/tex]

C. [tex]x=-15[/tex] or [tex]x=7[/tex]

D. [tex]x=-11[/tex] or [tex]x=11[/tex]

Answer :

Sure, let's solve the equation [tex]\( |x - 4| - 10 = 1 \)[/tex] step by step.

### Step 1: Isolate the absolute value expression
First, we need to isolate the absolute value part of the equation:
[tex]\[ |x - 4| - 10 = 1 \][/tex]

We add 10 to both sides to get:
[tex]\[ |x - 4| = 11 \][/tex]

### Step 2: Set up two cases for the absolute value
The absolute value [tex]\( |x - 4| = 11 \)[/tex] means that [tex]\( x - 4 \)[/tex] can be either 11 or -11. This gives us two cases:

Case 1:
[tex]\[ x - 4 = 11 \][/tex]

Case 2:
[tex]\[ x - 4 = -11 \][/tex]

### Step 3: Solve for [tex]\( x \)[/tex] in both cases

For Case 1:
[tex]\[ x - 4 = 11 \][/tex]
Add 4 to both sides:
[tex]\[ x = 15 \][/tex]

For Case 2:
[tex]\[ x - 4 = -11 \][/tex]
Add 4 to both sides:
[tex]\[ x = -7 \][/tex]

### Step 4: State the solutions
The solutions to the equation [tex]\( |x - 4| - 10 = 1 \)[/tex] are:
[tex]\[ x = 15 \text{ or } x = -7 \][/tex]

So, from the given options, the correct answer is:
[tex]\[ x = -7 \text{ or } x = 15 \][/tex]

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