Question
$$\frac{x}{2x-5}+\frac{1}{x+5} = \frac{18}{3x-5}$$
Answer
This equation has no solution.
Explanation
$$ \begin{aligned} \frac{x}{2x-5}+\frac{1}{x+5} &= \frac{18}{3x-5}&& \text{multiply ALL terms by } \color{blue}{ (2x-5)(x+5)(3x-5) }. \\[1 em](2x-5)(x+5)(3x-5)\frac{x}{2x-5}+(2x-5)(x+5)(3x-5)\cdot\frac{1}{x+5} &= (2x-5)(x+5)(3x-5)\cdot\frac{18}{3x-5}&& \text{cancel out the denominators} \\[1 em]3x^3+10x^2-25x+6x^2-25x+25 &= 36x^2+90x-450&& \text{simplify left side} \\[1 em]3x^3+16x^2-50x+25 &= 36x^2+90x-450&& \text{move all terms to the left hand side } \\[1 em]3x^3+16x^2-50x+25-36x^2-90x+450 &= 0&& \text{simplify left side} \\[1 em]3x^3-20x^2-140x+475 &= 0&& \\[1 em] \end{aligned} $$
This polynomial has no rational roots that can be found using Rational Root Test.
Roots were found using qubic formulas.
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Equations Solver