Certainly! Let's solve the absolute value equation [tex]\( |x - 7| = c \)[/tex] step-by-step.
### Step 1: Understand the Definition
The absolute value [tex]\( |a| \)[/tex] of a number [tex]\( a \)[/tex] is defined as:
[tex]\[ |a| =
\begin{cases}
a, & \text{if } a \geq 0 \\
-a, & \text{if } a < 0
\end{cases}
\][/tex]
In our equation [tex]\( |x - 7| = c \)[/tex], the expression inside the absolute value function is [tex]\( x - 7 \)[/tex].
### Step 2: Set Up the Cases
For the equation [tex]\( |x - 7| = c \)[/tex] to hold true, the expression [tex]\( x - 7 \)[/tex] must equal [tex]\( c \)[/tex] or [tex]\( -c \)[/tex].
Thus, we will split this into two separate cases:
1. [tex]\( x - 7 = c \)[/tex]
2. [tex]\( x - 7 = -c \)[/tex]
### Case 1: [tex]\( x - 7 = c \)[/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ x - 7 = c \][/tex]
Add 7 to both sides of the equation:
[tex]\[ x = c + 7 \][/tex]
### Case 2: [tex]\( x - 7 = -c \)[/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ x - 7 = -c \][/tex]
Add 7 to both sides of the equation:
[tex]\[ x = 7 - c \][/tex]
### Step 3: Combine the Solutions
The solutions to the absolute value equation [tex]\( |x - 7| = c \)[/tex] are:
[tex]\[ x = c + 7 \][/tex]
[tex]\[ x = 7 - c \][/tex]
### Step 4: Consider the Value of [tex]\( c \)[/tex]
It's important to note that for the original absolute value equation [tex]\( |x - 7| = c \)[/tex] to make sense, [tex]\( c \)[/tex] must be non-negative. In other words, [tex]\( c \geq 0 \)[/tex].
### Final Answer
Therefore, the solutions to the equation [tex]\( |x - 7| = c \)[/tex], where [tex]\( c \geq 0 \)[/tex], are:
[tex]\[ x = 7 + c \][/tex]
[tex]\[ x = 7 - c \][/tex]
These are the two values of [tex]\( x \)[/tex] that satisfy the given absolute value equation.
