Prove the given statement using your information. "If x is a positive integer or a solution to x+3>4, then x>0 and x>12."

To prove this statement , we can use mathematical induction . \n \n Base case : When x = 1 , the statement is true because 1 + 3 4 . \n \n Ind uct ive step : Assume that the statement is true for some integer k , where k 1 . That is , we have shown that if x = k is a positive integer or a solution to k + 3 4 , then x 0 and x 1 2 . \n \n We need to show that the statement is also true for x = k + 1 . If x = k + 1 is a positive integer or a solution to ( k + 1 ) + 3 4 , then we can write it as k + 3 4 . By the induct ive hypothesis , we know that k + 3 0 and k + 1 2 0 . Therefore , k + 1 0 and ( k + 1 ) + 3 4 . This means that x = k + 1 satisfies the statement . \n \n There fore , the statement is true for all positive integers x , where x + 3 4 . </s>