Question

Solve the equation for x: 2x + b = c
Check this process:
Step 1: 2x + b - b = c - b
Step 2: 2x = c - b
Step 3: 2x - 2 = c - b - 2
Step 4: x = c/2 - b - 2
Check all of the choices that are true statements.
A. There is no error in the process
B. The process has an error from step 1 to step 2
C. The process has an error from step 2 to step 3
D. The process has an error from step 3 to step 4

Answer 1

Let's evaluate each step of the process given for solving the equation \(2x + b = c\):

1. **Step 1**: \(2x + b - b = c - b\)
- This step is correct. When you subtract \(b\) from both sides, you maintain the equality.

2. **Step 2**: \(2x = c - b\)
- This step is also correct. The equation simplifies to \(2x = c - b\) after \(b\) is cancelled out.

3. **Step 3**: \(2x - 2 = c - b - 2\)
- This step contains an error. The correct move from \(2x = c - b\) should be to isolate \(x\); there is no justification for subtracting 2 from both sides here.

4. **Step 4**: \(x = \frac{c}{2} - b - 2\)
- Since the previous step was incorrect, this equation is also incorrect. The correct logic from \(2x = c - b\) is to divide both sides by 2 to get \(x = \frac{c - b}{2}\).

Now that we have analyzed each step, let's evaluate the choices:

- **A. There is no error in the process**: **False** (there is an error).
- **B. The process has an error from step 1 to step 2**: **False** (steps 1 and 2 are correct).
- **C. The process has an error from step 2 to step 3**: **True** (this is where the first error occurs).
- **D. The process has an error from step 3 to step 4**: **True** (because step 3 was incorrect, step 4 is also incorrect).

Thus, the true statements are:
**C and D.**