Question

Solve the inequality $\frac{x-2}{x}<\frac{x-4}{x-6}$. Check your solution.

Short answer

The solution for the given inequality is$0<x<3\text{or}x>6$.

Step-by-step solution

Step 1. Write down the given expression.

The given inequality is $\frac{x-2}{x}<\frac{x-4}{x-6}$.

Step 2. Simplify the equation.

The given inequality $\frac{x-2}{x}<\frac{x-4}{x-6}$is solved as:

$\begin{array}{l}\frac{x-2}{x}<\frac{x-4}{x-6}\mathrm{....}\left(\text{Given}\right)\\ \frac{x-2}{x}-\frac{x-4}{x-6}<0\\ \frac{\left(x-2\right)\left(x-6\right)-x\left(x-4\right)}{x\left(x-6\right)}<0\\ \frac{x^2-8x+12-x^2+4x}{x\left(x-6\right)}<0\\ \frac{-4x+12}{x\left(x-6\right)}<0\end{array}$

Further simplifying,

$\begin{array}{l}\frac{-4\left(x-3\right)}{x\left(x-6\right)}<0\\ \frac{\left(x-3\right)}{x\left(x-6\right)}>\mathrm{0....}\left(\begin{array}{l}\text{Divideby}-4\text{and}\\ \text{reversetheinequality}\end{array}\right)\\ 0<x<3\text{or}x>6\end{array}$

Step 3. Verify the result for obtained value.

Plugging $x>6$say $x=7$in given inequality and verify the result.

$\begin{array}{l}\frac{x-2}{x}<\frac{x-4}{x-6}\mathrm{....}\left(\text{Given}\right)\\ \frac{\left(7\right)-2}{\left(7\right)}<\frac{\left(7\right)-4}{\left(7\right)-6}\mathrm{....}\left(\text{Plugging}x=7\right)\\ \frac{5}{7}<\frac{3}{1}\\ \frac{5}{7}<\mathrm{3....}\left(\text{True}\right)\end{array}$

Hence the solution is verified for$x=7$.

Step 4. Conclusion.

The solution for the given inequality is $0<x<3\text{or}x>6$.