To determine which of the given true conditional statements has a true converse, let's first recall the definition of a converse. The converse of a conditional statement "If [tex]\( P \)[/tex], then [tex]\( Q \)[/tex]" is "If [tex]\( Q \)[/tex], then [tex]\( P \)[/tex]".
We will analyze each of the given true conditional statements and their converses.
1. Statement: If [tex]\( x > 3 \)[/tex], then [tex]\( x > 2 \)[/tex].
- Converse: If [tex]\( x > 2 \)[/tex], then [tex]\( x > 3 \)[/tex].
This converse is not always true because there are values of [tex]\( x \)[/tex] (for example, [tex]\( x = 2.5 \)[/tex]) that are greater than 2 but not greater than 3. Therefore, this converse is false.
2. Statement: If [tex]\( x > 3 \)[/tex], then [tex]\( 2x > 6 \)[/tex].
- Converse: If [tex]\( 2x > 6 \)[/tex], then [tex]\( x > 3 \)[/tex].
To check if this converse is true, we can solve the inequality [tex]\( 2x > 6 \)[/tex]. Dividing both sides by 2, we get [tex]\( x > 3 \)[/tex]. This is exactly the same as the original condition. Therefore, this converse is true.
3. Statement: If [tex]\( x > 3 \)[/tex], then [tex]\( x > -3 \)[/tex].
- Converse: If [tex]\( x > -3 \)[/tex], then [tex]\( x > 3 \)[/tex].
This converse is false because there are many values of [tex]\( x \)[/tex] (for example, [tex]\( x = 0 \)[/tex]) that are greater than -3 but not greater than 3.
4. Statement: If [tex]\( x > 3 \)[/tex], then [tex]\( x > 0 \)[/tex].
- Converse: If [tex]\( x > 0 \)[/tex], then [tex]\( x > 3 \)[/tex].
This converse is also false because there are values of [tex]\( x \)[/tex] (for example, [tex]\( x = 1 \)[/tex]) that are greater than 0 but not greater than 3.
Based on this analysis, the true conditional statement that has a true converse is the second statement:
If [tex]\( x > 3 \)[/tex], then [tex]\( 2x > 6 \)[/tex].
Thus, the correct answer is the second statement.
