Answer :
To solve the equation [tex]\( |x-4| + 6 = 17 \)[/tex], let's go through the steps:
1. Isolate the Absolute Value: First, we need to get the absolute value by itself. So, subtract 6 from both sides of the equation:
[tex]\[
|x-4| + 6 - 6 = 17 - 6
\][/tex]
[tex]\[
|x-4| = 11
\][/tex]
2. Set Up Two Separate Equations: The property of absolute value tells us that if [tex]\( |A| = B \)[/tex], then [tex]\( A = B \)[/tex] or [tex]\( A = -B \)[/tex]. Here, [tex]\( A \)[/tex] is [tex]\( x-4 \)[/tex] and [tex]\( B \)[/tex] is 11. This gives us two equations:
Case 1:
[tex]\[
x - 4 = 11
\][/tex]
Case 2:
[tex]\[
x - 4 = -11
\][/tex]
3. Solve Each Equation:
- For Case 1: [tex]\( x - 4 = 11 \)[/tex]
[tex]\[
x = 11 + 4 = 15
\][/tex]
- For Case 2: [tex]\( x - 4 = -11 \)[/tex]
[tex]\[
x = -11 + 4 = -7
\][/tex]
4. Solution:
The solutions are [tex]\( x = 15 \)[/tex] and [tex]\( x = -7 \)[/tex].
So, the correct answer is option A: [tex]\( x=15 \)[/tex] and [tex]\( x=-7 \)[/tex].
1. Isolate the Absolute Value: First, we need to get the absolute value by itself. So, subtract 6 from both sides of the equation:
[tex]\[
|x-4| + 6 - 6 = 17 - 6
\][/tex]
[tex]\[
|x-4| = 11
\][/tex]
2. Set Up Two Separate Equations: The property of absolute value tells us that if [tex]\( |A| = B \)[/tex], then [tex]\( A = B \)[/tex] or [tex]\( A = -B \)[/tex]. Here, [tex]\( A \)[/tex] is [tex]\( x-4 \)[/tex] and [tex]\( B \)[/tex] is 11. This gives us two equations:
Case 1:
[tex]\[
x - 4 = 11
\][/tex]
Case 2:
[tex]\[
x - 4 = -11
\][/tex]
3. Solve Each Equation:
- For Case 1: [tex]\( x - 4 = 11 \)[/tex]
[tex]\[
x = 11 + 4 = 15
\][/tex]
- For Case 2: [tex]\( x - 4 = -11 \)[/tex]
[tex]\[
x = -11 + 4 = -7
\][/tex]
4. Solution:
The solutions are [tex]\( x = 15 \)[/tex] and [tex]\( x = -7 \)[/tex].
So, the correct answer is option A: [tex]\( x=15 \)[/tex] and [tex]\( x=-7 \)[/tex].
