To write an absolute value equation of the form [tex]\( |x - b| = c \)[/tex] that has a solution [tex]\( x = -3 \)[/tex], follow these steps:
1. Understand the properties of absolute value:
- The expression [tex]\( |x - b| \)[/tex] represents the distance of [tex]\( x \)[/tex] from [tex]\( b \)[/tex].
- Therefore, [tex]\( |x - b| = c \)[/tex] means [tex]\( x \)[/tex] is [tex]\( c \)[/tex] units away from [tex]\( b \)[/tex].
2. Identify the given solution:
- One of the solutions is [tex]\( x = -3 \)[/tex].
- This means that when [tex]\( x = -3 \)[/tex], the absolute value expression [tex]\( |x - b| \)[/tex] should equal [tex]\( c \)[/tex].
3. Set up the absolute value equation:
- Let [tex]\( b \)[/tex] be the value such that [tex]\( x = -3 \)[/tex] is at a distance [tex]\( c \)[/tex] from [tex]\( b \)[/tex].
4. Determine the values of [tex]\( b \)[/tex] and [tex]\( c \)[/tex] based on the given solution:
- Since [tex]\( -3 \)[/tex] is a solution, we want [tex]\( |x - b| = c \)[/tex] to hold true when [tex]\( x = -3 \)[/tex].
- Substitute [tex]\( x = -3 \)[/tex] into [tex]\( |x - b| = c \)[/tex]:
[tex]\[
|-3 - b| = c
\][/tex]
- For [tex]\( |-3 - b| \)[/tex] to equal [tex]\( c \)[/tex], an obvious choice is [tex]\( b = 0 \)[/tex], making the equation simplify to:
[tex]\[
|-3 - 0| = 3 \Rightarrow c = 3
\][/tex]
5. Form the equation:
- With [tex]\( b = 0 \)[/tex] and [tex]\( c = 3 \)[/tex], the absolute value expression becomes:
[tex]\[
|x - 0| = 3 \quad \text{or simply} \quad |x| = 3
\][/tex]
Therefore, the absolute value equation that has [tex]\( x = -3 \)[/tex] as one of its solutions is:
[tex]\[
|x - 0| = 3 \quad \text{or} \quad |x| = 3
\][/tex]
This equation also has another solution, [tex]\( x = 3 \)[/tex], since both [tex]\( -3 \)[/tex] and [tex]\( 3 \)[/tex] are 3 units away from 0.
