Solve the inequality (x + 1)(x - 4) < 0.

Step-by-step solution

1. 1

   Read the critical values straight off the factors. The expression is already factored, so it vanishes at $x=-1$ and $x=4$. These two points cut the number line into three intervals: $(-\infty,-1)$, $(-1,4)$ and $(4,\infty)$. A product can only change sign where one of its factors is zero, so the sign is constant inside each interval.

2. 2

   Predict the answer from the shape. Expanding gives $x^2-3x-4$, an upward-opening parabola with roots $-1$ and $4$. Such a curve lies below the axis only between its roots, so we already expect $-1<x<4$; the interval tests below confirm it.

3. 3

   Test one convenient point per interval.

   $x=-2:\ (-1)(-6)=6>0,\qquad x=0:\ (1)(-4)=-4<0,\qquad x=5:\ (6)(1)=6>0$

   One point suffices per interval precisely because the sign cannot change inside it.

4. 4

   Select the interval where the product is negative. Only the middle interval qualifies:

   $-1<x<4$

   Structurally, this is where the two factors have opposite signs: $x+1>0$ while $x-4<0$.

5. 5

   Decide the endpoints. At $x=-1$ and $x=4$ the product equals $0$, and $0<0$ is false, so both endpoints are excluded. The solution set is the open interval

   $x\in(-1,4)$

   (Had the inequality been $\le 0$, the answer would be the closed interval $[-1,4]$.)