If `x = a + 2 and b = x + 1`, then which of the following must be false?

Options

A. `a gt b`
B. `a lt b`
C. `a = b`
D. `a = b^2`

Answer

To solve the problem, we start with the given equations:

1. \( x = a + 2 \)
2. \( b = x + 1 \)

### Step 1: Substitute \( x \) in the equation for \( b \)

We can substitute the expression for \( x \) from the first equation into the second equation.

\[ b = (a + 2) + 1 \]

### Step 2: Simplify the equation for \( b \)

Now, we simplify the equation:

\[ b = a + 2 + 1 = a + 3 \]

### Step 3: Analyze the relationship between \( a \) and \( b \)

From the equation \( b = a + 3 \), we can see that:

- \( b \) is always greater than \( a \) because \( 3 \) is a positive number.
- Therefore, we can conclude that \( b > a \).

### Step 4: Evaluate the options

Now we need to evaluate the options given in the question:

1. **Option A**: \( a > b \)
This must be false because \( b = a + 3 \), which means \( b \) is greater than \( a \).

2. **Option B**: \( a < b \)
This must be true because \( b = a + 3 \).

3. **Option C**: \( a = b \)
This must be false because \( b = a + 3 \) implies that \( a \) cannot equal \( b \).

4. **Option D**: \( a = b^2 \)
This must also be false because \( b = a + 3 \) does not allow \( a \) to equal \( b^2 \) for all values of \( a \).

### Conclusion

The statements that must be false are:

- \( a > b \) (Option A)
- \( a = b \) (Option C)
- \( a = b^2 \) (Option D)

Thus, the options that must be false are A, C, and D.