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Identify the biconditional of the following conditional statement.

Conditional: If [tex]$x+2=7$[/tex], then [tex]$x=5$[/tex].

A. There is no biconditional for these statements.
B. If [tex]$x+2=7$[/tex], then [tex]$x=5$[/tex].
C. [tex]$x=5$[/tex] if and only if [tex]$x+2=7$[/tex]; False.
D. [tex]$x+2=7$[/tex] if and only if [tex]$x=5$[/tex]; True.

Answer :

Let's analyze the given problem step-by-step to identify the bi-conditional statement for the given conditional statement.

### Conditional Statement:
"If [tex]\( x + 2 = 7 \)[/tex], then [tex]\( x = 5 \)[/tex]"

### Bi-Conditional Statement:
A bi-conditional statement combines a conditional statement and its converse. It states that both the original statement and its converse are true. It's written as "p if and only if q".

### Steps to Identify the Bi-Conditional:
1. Original Conditional:
- If [tex]\( x + 2 = 7 \)[/tex], then [tex]\( x = 5 \)[/tex].

2. Converse:
- If [tex]\( x = 5 \)[/tex], then [tex]\( x + 2 = 7 \)[/tex].

3. Combining These into a Bi-Conditional:
- [tex]\( x + 2 = 7 \)[/tex] if and only if [tex]\( x = 5 \)[/tex].

### Verifying the Truth of the Bi-Conditional Statement:
Let's verify the validity of this bi-conditional statement:

1. From the Original Conditional:
- If [tex]\( x + 2 = 7 \)[/tex] -> Solve for [tex]\( x \)[/tex]:
- [tex]\( x + 2 = 7 \)[/tex]
- [tex]\( x = 7 - 2 \)[/tex]
- [tex]\( x = 5 \)[/tex]

Hence, if [tex]\( x + 2 = 7 \)[/tex], [tex]\( x = 5 \)[/tex] is true.

2. From the Converse:
- If [tex]\( x = 5 \)[/tex] -> Substitute into the equation:
- [tex]\( x + 2 = 5 + 2 \)[/tex]
- [tex]\( x + 2 = 7 \)[/tex]

Hence, if [tex]\( x = 5 \)[/tex], [tex]\( x + 2 = 7 \)[/tex] is true as well.

Since both the original conditional and its converse are true, the bi-conditional statement [tex]\( x + 2 = 7 \)[/tex] if and only if [tex]\( x = 5 \)[/tex] is true.

### Conclusion:
The correct bi-conditional statement is:

Option (D): [tex]\( x + 2 = 7 \)[/tex] if and only if [tex]\( x = 5 \)[/tex]; True

Therefore, the answer is:
- Option (D): [tex]\( x + 2 = 7 \)[/tex] if and only if [tex]\( x = 5 \)[/tex]; True

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