Given: p: x – 5 =10 q: 4x + 1 = 61 which is the inverse of p → q? if x – 5 ≠ 10, then 4x + 1 ≠ 61. if 4x + 1 ≠ 61, then x – 5 ≠ 10. if x – 5 = 10, then 4x + 1 = 61. if 4x + 1 = 61, then x – 5 = 10.

If x – 5 ≠ 10, then 4x + 1 ≠ 61

Justification:

1) The inverse of a conditional is negating both the hipothesis and the conclusion of the conditional, keeping the same sense of the implication.

2) This is the scheme (the symbol ~ is used to negate)

conditional: p → q
hypothesis: p
conclusion: q

negated hypothesis: ~p
negated conclusion: ~ q

inverse conditional: ~p → ~q

3) So, for the hypotheis p: x – 5 =10 and the conclusion q: 4x + 1 = 61, the conditional p→ q is:

 if x - 5 = 10 then 4x + 1 = 61.

And the inverse is negating both the x - 5 = 10 and 4x + 1 = 61, leading to:

If x – 5 ≠ 10, then 4x + 1 ≠ 61, which is the answer.

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