To solve the equation [tex]\( |x - 11| = 3 \)[/tex], we need to understand what the absolute value operation means. The absolute value of a number represents its distance from zero on the number line, which is always nonnegative.
Given [tex]\( |x - 11| = 3 \)[/tex], this tells us that the distance between [tex]\( x \)[/tex] and 11 on the number line is 3. Therefore, there are two possible scenarios:
1. [tex]\( x - 11 = 3 \)[/tex]
2. [tex]\( x - 11 = -3 \)[/tex]
Let's solve each equation separately:
1. For [tex]\( x - 11 = 3 \)[/tex]:
[tex]\[
x - 11 = 3
\][/tex]
Add 11 to both sides:
[tex]\[
x = 3 + 11
\][/tex]
[tex]\[
x = 14
\][/tex]
2. For [tex]\( x - 11 = -3 \)[/tex]:
[tex]\[
x - 11 = -3
\][/tex]
Add 11 to both sides:
[tex]\[
x = -3 + 11
\][/tex]
[tex]\[
x = 8
\][/tex]
The solutions to the equation [tex]\( |x - 11| = 3 \)[/tex] are [tex]\( x = 14 \)[/tex] and [tex]\( x = 8 \)[/tex].
To find the positive difference between these two solutions, we calculate the absolute value of their difference:
[tex]\[
|14 - 8| = 6
\][/tex]
Thus, the positive difference between the two solutions for [tex]\( x \)[/tex] is:
[tex]\[
\boxed{6}
\][/tex]
