Let's solve the problem step by step.

The problem states:

• "The larger of two consecutive integers is 7 greater than twice the smaller."

Let the two consecutive integers be $x$ (the smaller integer) and $x + 1$ (the larger integer).

According to the problem, the larger integer is 7 greater than twice the smaller integer. This gives us the equation: $x + 1 = 2x + 7$

Solving the equation:

1. Subtract $x$ from both sides: $x + 1 = 2x + 7 \implies 1 = x + 7$

2. Subtract 7 from both sides: $1 - 7 = x \implies x = -6$

Now, since $x = -6$, the smaller integer is $-6$, and the larger integer is: $x + 1 = -6 + 1 = -5$

Conclusion:

The two consecutive integers are $-6$ and $-5$.

Would you like further details or explanations?

Here are 5 related questions to deepen your understanding:

1. What is the general form of a pair of consecutive integers?
2. How can we verify that these integers satisfy the given condition?
3. Can we modify the condition and solve for a different relationship between the integers?
4. What happens if the difference between the larger and twice the smaller is a negative number instead of 7?
5. How can this type of problem be generalized to non-consecutive numbers?

Tip: When dealing with consecutive numbers, always express one in terms of the other, like $x$ and $x + 1$, to simplify your equations.