Sure, let's derive the contrapositive of the given implication step-by-step.
### Step-by-Step Solution:
1. Identify the given statements:
- [tex]\( p: 3x + 7 = -5 \)[/tex]
- [tex]\( q: 4x + 12 = -4 \)[/tex]
2. Understand the logical implication:
- The given implication is [tex]\( p \rightarrow q \)[/tex], which reads "If [tex]\( p \)[/tex] (i.e., [tex]\( 3x + 7 = -5 \)[/tex]), then [tex]\( q \)[/tex] (i.e., [tex]\( 4x + 12 = -4 \)[/tex])."
3. Recall the contrapositive:
- The contrapositive of any statement [tex]\( p \rightarrow q \)[/tex] is [tex]\( \neg q \rightarrow \neg p \)[/tex].
- Here, [tex]\( \neg p \)[/tex] means "not [tex]\( p \)[/tex]" (i.e., [tex]\( 3x + 7 \neq -5 \)[/tex]).
- And [tex]\( \neg q \)[/tex] means "not [tex]\( q \)[/tex]" (i.e., [tex]\( 4x + 12 \neq -4 \)[/tex]).
4. Formulate the contrapositive:
- According to the definition above, the contrapositive of [tex]\( p \rightarrow q \)[/tex] ("If [tex]\( 3x + 7 = -5 \)[/tex], then [tex]\( 4x + 12 = -4 \)[/tex]") is:
- If [tex]\( \neg q \)[/tex] (i.e., [tex]\( 4x + 12 \neq -4 \)[/tex]), then [tex]\( \neg p \)[/tex] (i.e., [tex]\( 3x + 7 \neq -5 \)[/tex]).
5. State the result:
- Hence, the contrapositive of [tex]\( p \rightarrow q \)[/tex] is:
- "If [tex]\( 4x + 12 \neq -4 \)[/tex], then [tex]\( 3x + 7 \neq -5 \)[/tex]."
So, the correct contrapositive statement is:
- "If [tex]\( 4x + 12 \neq -4 \)[/tex], then [tex]\( 3x + 7 \neq -5 \)[/tex]."
Therefore, the right choice is:
- If [tex]\( 4x + 12 \neq -4 \)[/tex], then [tex]\( 3x + 7 \neq -5 \)[/tex].
