Question
Solve for all values of \( c \) in simplest form. \[ 1=|2 c+1| \]
Ask by Martin Cole. in the United States
Feb 19,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( c = -1 \) or \( c = 0 \).
Solution
Solve the equation \( 1=|2c+1| \).
Solve the equation by following steps:
- step0: Solve for \(c\):
\(1=\left|2c+1\right|\)
- step1: Swap the sides:
\(\left|2c+1\right|=1\)
- step2: Separate into possible cases:
\(\begin{align}&2c+1=1\\&2c+1=-1\end{align}\)
- step3: Solve the equation:
\(\begin{align}&c=0\\&c=-1\end{align}\)
- step4: Rewrite:
\(c_{1}=-1,c_{2}=0\)
The solutions for \( c \) in the simplest form are \( c = -1 \) and \( c = 0 \).
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Simplify this solution Bonus Knowledge
To solve the equation \( 1 = |2c + 1| \), we will consider the definition of absolute value. The expression \( |x| = a \) leads to two cases: \( x = a \) or \( x = -a \). 1. **Case 1:** \( 2c + 1 = 1 \) Subtracting 1 from both sides, we have: \[ 2c = 0 \] Dividing by 2 gives: \[ c = 0 \] 2. **Case 2:** \( 2c + 1 = -1 \) Subtracting 1 from both sides gives: \[ 2c = -2 \] Dividing by 2 results in: \[ c = -1 \] Thus, the solutions for \( c \) are \( c = 0 \) and \( c = -1 \). The final values of \( c \) in simplest form are: \[ \boxed{0 \text{ and } -1} \]
