To determine the converse of [tex]\( p \rightarrow q \)[/tex] given two statements [tex]\( p \)[/tex] and [tex]\( q \)[/tex]:
1. Identify the original statements:
- [tex]\( p \)[/tex]: [tex]\( 2x = 16 \)[/tex]
- [tex]\( q \)[/tex]: [tex]\( 3x - 4 = 20 \)[/tex]
2. Understand the converse:
- The converse of an implication [tex]\( p \rightarrow q \)[/tex] is [tex]\( q \rightarrow p \)[/tex].
3. Determine the solutions to the equations:
- For [tex]\( p \)[/tex]: [tex]\( 2x = 16 \)[/tex]
- Solving for [tex]\( x \)[/tex]: [tex]\( x = \frac{16}{2} = 8 \)[/tex]
- For [tex]\( q \)[/tex]: [tex]\( 3x - 4 = 20 \)[/tex]
- Solving for [tex]\( x \)[/tex]: [tex]\( x = \frac{20 + 4}{3} = 8 \)[/tex]
Both equations have the solution [tex]\( x = 8 \)[/tex].
4. Formulate the converse statement:
- The original implication [tex]\( p \rightarrow q \)[/tex] is: "If [tex]\( 2x = 16 \)[/tex], then [tex]\( 3x - 4 = 20 \)[/tex]".
- The converse [tex]\( q \rightarrow p \)[/tex] is: "If [tex]\( 3x - 4 = 20 \)[/tex], then [tex]\( 2x = 16 \)[/tex]".
5. Check the options:
- Option 1: If [tex]\( 2x \neq 16 \)[/tex], then [tex]\( 3x - 4 \neq 20 \)[/tex]. (This is not the correct converse)
- Option 2: If [tex]\( 3x - 4 \neq 20 \)[/tex], then [tex]\( 2x \neq 16 \)[/tex]. (This is not the correct converse)
- Option 3: If [tex]\( 2x = 16 \)[/tex], then [tex]\( 3x - 4 = 20 \)[/tex]. (This is the original statement [tex]\( p \rightarrow q \)[/tex])
- Option 4: If [tex]\( 3x - 4 = 20 \)[/tex], then [tex]\( 2x = 16 \)[/tex]. (This is the correct converse [tex]\( q \rightarrow p \)[/tex])
Therefore, the correct answer is:
If [tex]\( 3x - 4 = 20 \)[/tex], then [tex]\( 2x = 16 \)[/tex].
