To determine the inverse of the statement [tex]\( p \rightarrow q \)[/tex], it is essential to use logical reasoning and analyze the given mathematical statements.
First, let's understand the logical structures [tex]\( p \rightarrow q \)[/tex] and its inverse.
- [tex]\( p \rightarrow q \)[/tex] means "If [tex]\( p \)[/tex] is true, then [tex]\( q \)[/tex] is true."
- The inverse of [tex]\( p \rightarrow q \)[/tex] is [tex]\( \neg q \rightarrow \neg p \)[/tex], which means "If [tex]\( q \)[/tex] is false, then [tex]\( p \)[/tex] is false."
Given:
- [tex]\( p: x - 5 = 10 \)[/tex]
- [tex]\( q: 4x + 1 = 61 \)[/tex]
Let's solve for [tex]\( x \)[/tex] in both equations:
1. Solving [tex]\( p: x - 5 = 10 \)[/tex]:
[tex]\[
x - 5 = 10 \implies x = 10 + 5 \implies x = 15
\][/tex]
2. Solving [tex]\( q: 4x + 1 = 61 \)[/tex]:
[tex]\[
4x + 1 = 61 \implies 4x = 61 - 1 \implies 4x = 60 \implies x = \frac{60}{4} \implies x = 15
\][/tex]
Both equations have the solution [tex]\( x = 15 \)[/tex].
Now, to find the inverse of [tex]\( p \rightarrow q \)[/tex]:
- [tex]\( p \rightarrow q \)[/tex] means "If [tex]\( x - 5 = 10 \)[/tex], then [tex]\( 4x + 1 = 61 \)[/tex]."
The inverse [tex]\( \neg q \rightarrow \neg p \)[/tex] would be:
- [tex]\( \neg q \rightarrow \neg p \)[/tex] means "If [tex]\( q \)[/tex] is false, then [tex]\( p \)[/tex] is false."
- [tex]\( \neg q \)[/tex] is "If [tex]\( 4x + 1 \neq 61 \)[/tex]."
- [tex]\( \neg p \)[/tex] is "If [tex]\( x - 5 \neq 10 \)[/tex]."
Therefore, 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 statement is:
"If [tex]\( 4x + 1 \neq 61 \)[/tex], then [tex]\( x - 5 \neq 10 \)[/tex]."
