Answer :
Sure, let's solve the equation [tex]\( |x - 4| + 6 = 17 \)[/tex] step by step.
### Step 1: Isolate the Absolute Value Expression
First, subtract 6 from both sides of the equation to isolate the absolute value expression:
[tex]\[ |x - 4| + 6 - 6 = 17 - 6 \][/tex]
[tex]\[ |x - 4| = 11 \][/tex]
### Step 2: Set Up Two Cases
Since the absolute value [tex]\( |x - 4| \)[/tex] can be either [tex]\( 11 \)[/tex] or [tex]\( -11 \)[/tex] (although absolute values are always non-negative, we consider the negative counterpart for solving):
1. [tex]\( x - 4 = 11 \)[/tex]
2. [tex]\( x - 4 = -11 \)[/tex]
### Step 3: Solve Each Case Separately
#### Case 1: [tex]\( x - 4 = 11 \)[/tex]
Add 4 to both sides:
[tex]\[ x = 11 + 4 \][/tex]
[tex]\[ x = 15 \][/tex]
#### Case 2: [tex]\( x - 4 = -11 \)[/tex]
Add 4 to both sides:
[tex]\[ x = -11 + 4 \][/tex]
[tex]\[ x = -7 \][/tex]
### Step 4: Combine Solutions
The solutions to the equation [tex]\( |x - 4| + 6 = 17 \)[/tex] are:
[tex]\[ x = 15 \text{ and } x = -7 \][/tex]
### Step 5: Choose the Correct Answer
From the given options:
A. [tex]\( x = -15 \)[/tex] and [tex]\( x = 7 \)[/tex]
B. [tex]\( x = 15 \)[/tex] and [tex]\( x = -7 \)[/tex]
C. [tex]\( x = 15 \)[/tex] and [tex]\( x = -15 \)[/tex]
D. [tex]\( x = -15 \)[/tex] and [tex]\( x = -7 \)[/tex]
The correct answer is:
B. [tex]\( x = 15 \)[/tex] and [tex]\( x = -7 \)[/tex]
So, option B is the correct solution.
### Step 1: Isolate the Absolute Value Expression
First, subtract 6 from both sides of the equation to isolate the absolute value expression:
[tex]\[ |x - 4| + 6 - 6 = 17 - 6 \][/tex]
[tex]\[ |x - 4| = 11 \][/tex]
### Step 2: Set Up Two Cases
Since the absolute value [tex]\( |x - 4| \)[/tex] can be either [tex]\( 11 \)[/tex] or [tex]\( -11 \)[/tex] (although absolute values are always non-negative, we consider the negative counterpart for solving):
1. [tex]\( x - 4 = 11 \)[/tex]
2. [tex]\( x - 4 = -11 \)[/tex]
### Step 3: Solve Each Case Separately
#### Case 1: [tex]\( x - 4 = 11 \)[/tex]
Add 4 to both sides:
[tex]\[ x = 11 + 4 \][/tex]
[tex]\[ x = 15 \][/tex]
#### Case 2: [tex]\( x - 4 = -11 \)[/tex]
Add 4 to both sides:
[tex]\[ x = -11 + 4 \][/tex]
[tex]\[ x = -7 \][/tex]
### Step 4: Combine Solutions
The solutions to the equation [tex]\( |x - 4| + 6 = 17 \)[/tex] are:
[tex]\[ x = 15 \text{ and } x = -7 \][/tex]
### Step 5: Choose the Correct Answer
From the given options:
A. [tex]\( x = -15 \)[/tex] and [tex]\( x = 7 \)[/tex]
B. [tex]\( x = 15 \)[/tex] and [tex]\( x = -7 \)[/tex]
C. [tex]\( x = 15 \)[/tex] and [tex]\( x = -15 \)[/tex]
D. [tex]\( x = -15 \)[/tex] and [tex]\( x = -7 \)[/tex]
The correct answer is:
B. [tex]\( x = 15 \)[/tex] and [tex]\( x = -7 \)[/tex]
So, option B is the correct solution.
