If x < 0, we need to consider the sign change when dividing by a negative number.

When dividing by a negative number, the inequality sign must be reversed. Therefore, if x < 0, we have:

x/6 > -x/5

To compare the two fractions, we can multiply both sides of the inequality by a positive value to maintain the inequality:

5x/6 > -x/5

Now, let's simplify the inequality:

25x > -6x

Adding 6x to both sides:

31x > 0

Since x < 0, and multiplying a negative number by a positive number results in a negative number, the inequality 31x > 0 is false when x < 0.

To summarize, if x < 0, the inequality x/6 < -x/5 is false.