1. **State the problem:** We need to find two consecutive even integers such that five times the smaller integer is less than four times the greater integer. 2. **Define variables:** Let the smaller even integer be $x$. Since the integers are consecutive even numbers, the next even integer (the greater) is $x+2$. 3. **Set up the inequality:** According to the problem, five times the smaller is less than four times the greater: $$5x < 4(x + 2)$$ 4. **Simplify the inequality:** $$5x < 4x + 8$$ 5. **Solve for $x$:** Subtract $4x$ from both sides: $$5x - 4x < 8$$ $$x < 8$$ 6. **Interpret the inequality:** Since $x$ is an even integer less than 8, possible candidates for $x$ are even integers less than 8, for example, ..., -4, -2, 0, 2, 4, 6. 7. **Find the largest pair:** The largest even integer less than 8 is 6. So, the smaller integer is $6$ and the larger integer is $6 + 2 = 8$. 8. **Verify the inequality for this pair:** $$5 \times 6 = 30$$ $$4 \times 8 = 32$$ Since $$30 < 32$$, the condition is satisfied. **Final answer:** The largest pair of consecutive even integers satisfying the condition is $6$ and $8$.