Answer
$(1, 10)$
Work Step by Step
1. Express the inequality in the form $f(x)<0, f(x)>0, f(x)\leq 0$, or $f(x)\geq 0,$
where $f$ is a Rational function.
$\displaystyle \frac{x-1}{x-10} \lt 0$
$f(x)=\displaystyle \frac{x-1}{x-10}$
2. The cut points are:
$\displaystyle \frac{x-1}{x-10} \lt 0$
$x=1$ or $x=10$
3. Make a table or diagram: use the test values to make a table or diagram of the sign of each factor in each interval.
4. Test each interval's sign of $f(x)$ with a test value,
$\begin{array}{llll}
Intervals: & a=test.v. & f(a),signs & f(a) \lt 0 ? \\
& & \frac{a-1}{a-10} & \\
(-\infty,1) & 0 & \frac{(-)}{(-)}=(+) & F\\
(1,10) & 5 & \frac{(+)}{(-)}=(-) & T\\
(10,\infty) & 15 & \frac{(+)}{(+)}=(+) & F\\
\end{array}$
5. Write the solution set, selecting the interval or intervals that satisfy the given inequality. If the inequality involves $\leq$ or $\geq$, include the boundary points.
Solution set: $(1, 10)$
