# How do you solve (x-4)(x+3)<0 using a sign chart?

Jan 2, 2017

$- 3 < x < 4$.

#### Explanation:

Find the critical points. Equate the "factors" to 0 to get the critical points. Use 0 as a critical point also.

$x - 4 = 0 \text{ "=>" } x = 4$

$x + 3 = 0 \text{ "=>" "x = "-} 3$

The critical points are $\left\{\text{-} 3 , 0 , 4\right\}$. It is only at these points that the sign of $\left(x - 3\right) \left(x + 4\right)$ may change.

Now, pick a number that lies in each region (in-between/on either side of these critical points), plug it into each factor, and find the sign of each factor in each region. You'll make a table like this one.

$\underline{\text{ "x < "-"3" -"3 < x < 0" "0 < x < 4" "x > 4" }}$
$x + 3 \text{ "-" "+" "+" } +$
$\underline{x - 4 \text{ "-" "-" "-" "+" }}$
$\left(x - 4\right) \left(x + 3\right) \text{ "+" "-" "-" } +$

The last row is just the product of the two rows above it.

Also, since we manually added 0 as a critical point, we calculate $\left(x - 4\right) \left(x + 3\right)$ when $x = 0$:

$\left(0 - 4\right) \left(0 + 3\right) \text{ "=" "("-"4)(3)" "=" ""-"12" "<" } 0$

Based on this and the chart above,

$\left(x + 3\right) \left(x - 4\right) < 0$ when $- 3 < x < 4$.