To identify which of the following statements is a biconditional statement, we need to consider what a biconditional statement means. A biconditional statement is true in both directions—it means "if and only if." This implies that both conditions must be true simultaneously for the statement to hold.
Let's analyze each option:
1. Option A: If [tex]\( x \neq 5 \)[/tex], then [tex]\( x^2 \neq 25 \)[/tex]
This is a conditional statement but not biconditional. While it's true that if [tex]\( x \neq 5 \)[/tex], then [tex]\( x^2 \)[/tex] is likely not 25, [tex]\( x \)[/tex] could also be -5, violating the biconditional condition.
2. Option B: If [tex]\( x^2 = 25 \)[/tex], then [tex]\( x=5 \)[/tex] or [tex]\( x=-5 \)[/tex]
This statement is true but only in one direction. It tells us the possible values of [tex]\( x \)[/tex] when [tex]\( x^2 = 25 \)[/tex]; however, it does not work the other way to form a biconditional statement. It does not ensure if [tex]\( x=5 \)[/tex] or [tex]\( x=-5 \)[/tex], then [tex]\( x^2 = 25 \)[/tex] necessarily.
3. Option C: [tex]\( x=5 \)[/tex] if [tex]\( x^2=25 \)[/tex]
Similar to option B, this statement is also true in one direction. It means [tex]\( x=5 \)[/tex] if [tex]\( x^2 = 25 \)[/tex], but it’s not comprehensive because [tex]\( x^2 = 25 \)[/tex] also includes [tex]\( x = -5 \)[/tex], which is not covered here.
4. Option D: [tex]\( x=5 \)[/tex] if and only if [tex]\( x+5=10 \)[/tex]
This is a biconditional statement, as it holds true in both directions:
- If [tex]\( x=5 \)[/tex], then [tex]\( x+5=10 \)[/tex].
- If [tex]\( x+5=10 \)[/tex], then solving the equation [tex]\( x=5 \)[/tex].
After careful analysis, we determine that the correct answer is Option D, as it satisfies the biconditional requirement. Thus, the statement "x=5 if and only if x+5=10" is correct in both directions.
Therefore, the correct answer is:
Option D: [tex]\( x=5 \)[/tex] if and only if [tex]\( x+5 = 10 \)[/tex].
