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Which of these true conditional statements has a true converse?

A. If [tex]\( x \ \textgreater \ 3 \)[/tex], then [tex]\( x \ \textgreater \ 2 \)[/tex].
B. If [tex]\( x \ \textgreater \ 3 \)[/tex], then [tex]\( 2x \ \textgreater \ 6 \)[/tex].
C. If [tex]\( x \ \textgreater \ 3 \)[/tex], then [tex]\( x \ \textgreater \ -3 \)[/tex].
D. If [tex]\( x \ \textgreater \ 3 \)[/tex], then [tex]\( x \ \textgreater \ 0 \)[/tex].

Sagot :

GinnyAnswer
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.
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