To solve this problem, we'll analyze the contrapositive of the given logical implication [tex]\( p \rightarrow q \)[/tex]. Let's break the steps down systematically.
1. Identify the Propositions:
- [tex]\( p: 3x + 7 = -5 \)[/tex]
- [tex]\( q: 4x + 12 = -4 \)[/tex]
2. Understand the Implication [tex]\( p \rightarrow q \)[/tex]:
- "If [tex]\( 3x + 7 = -5 \)[/tex], then [tex]\( 4x + 12 = -4 \)[/tex]."
3. Formulate the Contrapositive:
- The contrapositive of [tex]\( p \rightarrow q \)[/tex] is [tex]\( \lnot q \rightarrow \lnot p \)[/tex], where [tex]\( \lnot q \)[/tex] means "not [tex]\( q \)[/tex]" and [tex]\( \lnot p \)[/tex] means "not [tex]\( p \)[/tex]".
4. Determine the Negations:
- [tex]\( \lnot p \)[/tex]: The negation of [tex]\( p: 3x + 7 = -5 \)[/tex] is [tex]\( 3x + 7 \neq -5 \)[/tex].
- [tex]\( \lnot q \)[/tex]: The negation of [tex]\( q: 4x + 12 = -4 \)[/tex] is [tex]\( 4x + 12 \neq -4 \)[/tex].
5. Construct the Contrapositive Statement:
- Based on the negations, the contrapositive [tex]\( \lnot q \rightarrow \lnot p \)[/tex] becomes:
- "If [tex]\( 4x + 12 \neq -4 \)[/tex], then [tex]\( 3x + 7 \neq -5 \)[/tex]."
Therefore, the correct answer is:
If [tex]\( 4x + 12 \neq -4 \)[/tex], then [tex]\( 3x + 7 \neq -5 \)[/tex].
