Let's carefully explore the problem and find the inverse of the given logical statement [tex]\( p \rightarrow q \)[/tex].
### Definitions and Given Statements
We have:
1. [tex]\( p: x - 5 = 10 \)[/tex]
2. [tex]\( q: 4x + 1 = 61 \)[/tex]
### Logical Implication
The statement [tex]\( p \rightarrow q \)[/tex] means "If [tex]\( p \)[/tex] is true, then [tex]\( q \)[/tex] is true."
### Inverse of [tex]\( p \rightarrow q \)[/tex]
The inverse of a conditional statement [tex]\( p \rightarrow q \)[/tex] is [tex]\( \neg q \rightarrow \neg p \)[/tex], where [tex]\( \neg \)[/tex] denotes the negation.
This means:
- [tex]\( \neg p \)[/tex]: the negation of [tex]\( p \)[/tex]: [tex]\( x - 5 \neq 10 \)[/tex]
- [tex]\( \neg q \)[/tex]: the negation of [tex]\( q \)[/tex]: [tex]\( 4x + 1 \neq 61 \)[/tex]
By the inverse definition, [tex]\( \neg q \rightarrow \neg p \)[/tex] translates to:
- "If [tex]\( q \)[/tex] is not true, then [tex]\( p \)[/tex] is not true."
Putting this together, the inverse of [tex]\( p \rightarrow q \)[/tex] is:
- "If [tex]\( 4x + 1 \neq 61 \)[/tex], then [tex]\( x - 5 \neq 10 \)[/tex]."
Thus, the correct inverse of [tex]\( p \rightarrow q \)[/tex] is:
- If [tex]\( 4 x + 1 \neq 61 \)[/tex], then [tex]\( x - 5 \neq 10 \)[/tex].
So, the correct answer is:
If [tex]\( 4 x + 1 \neq 61 \)[/tex], then [tex]\( x-5 \neq 10 \)[/tex].
