Certainly! Let's address the problem step-by-step to identify the converse of the implication [tex]\( p \rightarrow q \)[/tex].
### Definitions:
- Statement [tex]\( p \)[/tex]: [tex]\(2x = 16\)[/tex]
- Statement [tex]\( q \)[/tex]: [tex]\(3x - 4 = 20\)[/tex]
- Implication [tex]\( p \rightarrow q \)[/tex]: "If [tex]\(2x = 16\)[/tex], then [tex]\(3x - 4 = 20\)[/tex]."
### Analysis:
1. Original Implication ( [tex]\( p \rightarrow q \)[/tex] ):
- Meaning: If the equation [tex]\(2x = 16\)[/tex] is true, then the equation [tex]\(3x - 4 = 20\)[/tex] must also be true.
2. Converse of an Implication:
- The converse of an implication [tex]\( p \rightarrow q \)[/tex] is [tex]\( q \rightarrow p \)[/tex].
- Meaning: "If [tex]\(3x - 4 = 20\)[/tex], then [tex]\(2x = 16\)[/tex]."
### Solutions for the Original Implication and Its Converse:
Let's match our analysis with the given options:
1. Option 1: If [tex]\(2x \neq 16\)[/tex], then [tex]\(3x - 4 \neq 20\)[/tex].
- This is negation of both conditions.
2. Option 2: If [tex]\(3x - 4 \neq 20\)[/tex], then [tex]\(2x \neq 16\)[/tex].
- This also represents a negation but in reverse order.
3. Option 3: If [tex]\(2x = 16\)[/tex], then [tex]\(3x - 4 = 20\)[/tex].
- This is the original implication.
4. Option 4: If [tex]\(3x - 4 = 20\)[/tex], then [tex]\(2x = 16\)[/tex].
- This is the converse of the original implication.
### Conclusion:
The converse of the given implication [tex]\( p \rightarrow q \)[/tex] ("If [tex]\(2x = 16\)[/tex], then [tex]\(3x - 4 = 20\)[/tex]") is:
If [tex]\(3x - 4 = 20\)[/tex], then [tex]\(2x = 16\)[/tex].
Therefore, the correct answer is option 4.
