Solve (x + 25)/(x + 125) - x/(x + 100) = 1/10, stating any restrictions on x.

Step-by-step solution

1. 1

   Write down the restrictions. Neither denominator may vanish, so

   $x\ne-125,\qquad x\ne-100.$

   Both candidate roots will have to be checked against these at the end.

2. 2

   Combine over the common denominator and watch the collapse. Using $(x+125)(x+100)$:

   $(x+25)(x+100)=x^{2}+125x+2500,\qquad x(x+125)=x^{2}+125x,$

   so the numerator is

   $\left(x^{2}+125x+2500\right)-\left(x^{2}+125x\right)=2500.$

   Everything with an $x$ cancels, leaving a constant over a quadratic:

   $\frac{2500}{(x+125)(x+100)}=\frac{1}{10}.$

   Note the sign: subtracting in this order leaves $+2500$, whereas reversing the two fractions would leave $-2500$ — and that sign is what decides whether real solutions exist at all.

3. 3

   Cross-multiply and form the quadratic. From $25000=(x+125)(x+100)=x^{2}+225x+12500$:

   $x^{2}+225x-12500=0.$

   The constant is now negative, so by Vieta the two roots have opposite signs — one positive and one negative.

4. 4

   Solve and simplify the surd. The discriminant is

   $\Delta=225^{2}+4(12500)=50625+50000=100625>0,$

   so there are two real roots. Factoring out squares, $100625=625\times161$ and $161=7\times23$ is square-free, giving $\sqrt{100625}=25\sqrt{161}$. Hence

   $x=\frac{-225\pm25\sqrt{161}}{2}.$

5. 5

   Approximate and verify. With $\sqrt{161}\approx12.6886$:

   $x\approx46.107\qquad\text{or}\qquad x\approx-271.107.$

   Neither equals $-100$ or $-125$, so both are valid. Substituting $x=46.107$: $\frac{71.107}{171.107}-\frac{46.107}{146.107}=0.41557-0.31557=0.10000$ ✓, and the negative root checks out to the same precision ✓. Contrast with the reversed equation $\frac{x}{x+100}-\frac{x+25}{x+125}=\frac1{10}$, whose discriminant is $-99375$ and which has no real solution at all.