Answer :
To determine the converse of the statement [tex]\( p \rightarrow q \)[/tex], let's first review what each variable and implication represents.
### Original Statements:
- [tex]\( p: 2x = 16 \)[/tex]
- [tex]\( q: 3x - 4 = 20 \)[/tex]
- [tex]\( p \rightarrow q \)[/tex]: If [tex]\( 2x = 16 \)[/tex], then [tex]\( 3x - 4 = 20 \)[/tex].
### Converse Definition:
The converse of an implication [tex]\( p \rightarrow q \)[/tex] is [tex]\( q \rightarrow p \)[/tex].
### Analyze the Converse Configuration:
Starting with the given implication [tex]\( p \rightarrow q \)[/tex], where [tex]\( p \)[/tex] is [tex]\( 2x = 16 \)[/tex] and [tex]\( q \)[/tex] is [tex]\( 3x - 4 = 20 \)[/tex]:
- Original implication: If [tex]\( 2x = 16 \)[/tex], then [tex]\( 3x - 4 = 20 \)[/tex].
- Converse: If [tex]\( 3x - 4 = 20 \)[/tex], then [tex]\( 2x = 16 \)[/tex].
### Options Analysis:
Let's go through the choices one by one to identify which matches the converse [tex]\( q \rightarrow p \)[/tex].
1. If [tex]\( 2x \neq 16 \)[/tex], then [tex]\( 3x - 4 \neq 20 \)[/tex].
- This represents the contrapositive of the original statement, not the converse. The contrapositive of [tex]\( p \rightarrow q \)[/tex] is [tex]\( \neg q \rightarrow \neg p \)[/tex].
2. If [tex]\( 3x - 4 \neq 20 \)[/tex], then [tex]\( 2x \neq 16 \)[/tex].
- Similar to the first option, this is the contrapositive of the converse, which is [tex]\( \neg p \rightarrow \neg q \)[/tex] for the statement [tex]\( q \rightarrow p \)[/tex].
3. If [tex]\( 2x = 16 \)[/tex], then [tex]\( 3x - 4 = 20 \)[/tex].
- This is restating the original implication and not the converse.
4. If [tex]\( 3x - 4 = 20 \)[/tex], then [tex]\( 2x = 16 \)[/tex].
- This matches exactly with the converse statement [tex]\( q \rightarrow p \)[/tex].
### Conclusion:
Therefore, the converse of [tex]\( p \rightarrow q \)[/tex] is:
[tex]$\boxed{\text{If } 3x - 4 = 20, \text{ then } 2x = 16.}$[/tex]
### Original Statements:
- [tex]\( p: 2x = 16 \)[/tex]
- [tex]\( q: 3x - 4 = 20 \)[/tex]
- [tex]\( p \rightarrow q \)[/tex]: If [tex]\( 2x = 16 \)[/tex], then [tex]\( 3x - 4 = 20 \)[/tex].
### Converse Definition:
The converse of an implication [tex]\( p \rightarrow q \)[/tex] is [tex]\( q \rightarrow p \)[/tex].
### Analyze the Converse Configuration:
Starting with the given implication [tex]\( p \rightarrow q \)[/tex], where [tex]\( p \)[/tex] is [tex]\( 2x = 16 \)[/tex] and [tex]\( q \)[/tex] is [tex]\( 3x - 4 = 20 \)[/tex]:
- Original implication: If [tex]\( 2x = 16 \)[/tex], then [tex]\( 3x - 4 = 20 \)[/tex].
- Converse: If [tex]\( 3x - 4 = 20 \)[/tex], then [tex]\( 2x = 16 \)[/tex].
### Options Analysis:
Let's go through the choices one by one to identify which matches the converse [tex]\( q \rightarrow p \)[/tex].
1. If [tex]\( 2x \neq 16 \)[/tex], then [tex]\( 3x - 4 \neq 20 \)[/tex].
- This represents the contrapositive of the original statement, not the converse. The contrapositive of [tex]\( p \rightarrow q \)[/tex] is [tex]\( \neg q \rightarrow \neg p \)[/tex].
2. If [tex]\( 3x - 4 \neq 20 \)[/tex], then [tex]\( 2x \neq 16 \)[/tex].
- Similar to the first option, this is the contrapositive of the converse, which is [tex]\( \neg p \rightarrow \neg q \)[/tex] for the statement [tex]\( q \rightarrow p \)[/tex].
3. If [tex]\( 2x = 16 \)[/tex], then [tex]\( 3x - 4 = 20 \)[/tex].
- This is restating the original implication and not the converse.
4. If [tex]\( 3x - 4 = 20 \)[/tex], then [tex]\( 2x = 16 \)[/tex].
- This matches exactly with the converse statement [tex]\( q \rightarrow p \)[/tex].
### Conclusion:
Therefore, the converse of [tex]\( p \rightarrow q \)[/tex] is:
[tex]$\boxed{\text{If } 3x - 4 = 20, \text{ then } 2x = 16.}$[/tex]
