Solve for x
x\in (-\infty,\frac{1}{5})\cup [5,\infty)
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x-5\leq 0 5x-1<0
For the quotient to be ≥0, x-5 and 5x-1 have to be both ≤0 or both ≥0, and 5x-1 cannot be zero. Consider the case when x-5\leq 0 and 5x-1 is negative.
x<\frac{1}{5}
The solution satisfying both inequalities is x<\frac{1}{5}.
x-5\geq 0 5x-1>0
Consider the case when x-5\geq 0 and 5x-1 is positive.
x\geq 5
The solution satisfying both inequalities is x\geq 5.
x<\frac{1}{5}\text{; }x\geq 5
The final solution is the union of the obtained solutions.
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