To solve the equation [tex]\( |x - 2| = 6 \)[/tex], we need to consider the definition of absolute value. The absolute value equation [tex]\( |a| = b \)[/tex] implies two possible equations: [tex]\( a = b \)[/tex] or [tex]\( a = -b \)[/tex].
So, for the equation [tex]\( |x - 2| = 6 \)[/tex]:
1. We set [tex]\( x - 2 \)[/tex] equal to 6:
[tex]\[
x - 2 = 6
\][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[
x = 6 + 2
\][/tex]
[tex]\[
x = 8
\][/tex]
2. We set [tex]\( x - 2 \)[/tex] equal to -6:
[tex]\[
x - 2 = -6
\][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[
x = -6 + 2
\][/tex]
[tex]\[
x = -4
\][/tex]
Thus, the solutions to the equation [tex]\( |x - 2| = 6 \)[/tex] are [tex]\( x = 8 \)[/tex] and [tex]\( x = -4 \)[/tex].
To represent these solutions on the number line:
1. Draw a horizontal line and mark important points, particularly [tex]\( -4 \)[/tex] and [tex]\( 8 \)[/tex].
2. Place a point at [tex]\( -4 \)[/tex] and a point at [tex]\( 8 \)[/tex].
These two points, [tex]\( -4 \)[/tex] and [tex]\( 8 \)[/tex], are the solutions to the equation [tex]\( |x - 2| = 6 \)[/tex].
