Solve the inequality 3/(x - 1) > x + 4.

Step-by-step solution

1. 1

   Do not multiply both sides by $x-1$. Its sign is unknown — positive when $x>1$, negative when $x<1$ — so multiplying would flip the inequality on one half of the line and not the other. The safe route is to move everything to one side and compare a single fraction against zero:

   $\frac{3}{x-1}-(x+4)>0$

2. 2

   Combine over the common denominator. Expanding $(x+4)(x-1)=x^{2}+3x-4$:

   $\frac{3-(x^{2}+3x-4)}{x-1}=\frac{-x^{2}-3x+7}{x-1}>0$

   Multiplying numerator and denominator by $-1$ makes the leading coefficient positive, which reverses the inequality symbol:

   $\frac{x^{2}+3x-7}{x-1}<0$

3. 3

   Locate every critical point — zeros of the numerator and of the denominator. For the numerator, the quadratic formula gives

   $x=\frac{-3\pm\sqrt{9+28}}{2}=\frac{-3\pm\sqrt{37}}{2}\approx-4.5414,\;1.5414$

   and the denominator vanishes at $x=1$. The three points $\frac{-3-\sqrt{37}}{2}$, $1$, $\frac{-3+\sqrt{37}}{2}$ split the line into four intervals. Note $1$ lies between the two roots, so it genuinely interleaves them.

4. 4

   Build the sign chart. The numerator (upward parabola) is negative strictly between its roots; the denominator is negative for $x<1$. A quotient is negative when exactly one part is negative:

   $\begin{array}{c|c|c|c}\text{interval}&\text{num}&\text{den}&\text{quotient}\\\hline x<-4.5414&+&-&-\;\checkmark\\ -4.5414<x<1&-&-&+\\ 1<x<1.5414&-&+&-\;\checkmark\\ x>1.5414&+&+&+\end{array}$

5. 5

   Read off the solution and check the excluded point.

   $\left(-\infty,\tfrac{-3-\sqrt{37}}{2}\right)\cup\left(1,\tfrac{-3+\sqrt{37}}{2}\right)$

   All endpoints are open: the two roots give equality (not strict), and $x=1$ is excluded entirely because the original fraction is undefined there. Sampling the original inequality at $8000$ points across $[-10,6]$ agrees with this set everywhere ✓. Spot check $x=2$: $\frac{3}{1}=3$ versus $6$, and $3>6$ is false ✗, correctly outside.