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

1 Answer
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" "=>" "x = 4

x + 3 = 0" "=>" "x = "-"3

The critical points are {"-"3, 0, 4}. It is only at these points that the sign of (x-3)(x+4) 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.

ul("                         "x < "-"3"      -"3 < x < 0"       "0 < x < 4"       "x > 4"   ")
x + 3"                  "-"                 "+"                    "+"                "+
ul(x - 4"                  "-"                 "-"                    "-"                "+"     ")
(x - 4)(x + 3)"  "+"                 "-"                    "-"                "+

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 (x-4)(x+3) when x=0:

(0-4)(0+3)" "=" "("-"4)(3)" "=" ""-"12" "<" "0

Based on this and the chart above,

(x+3)(x-4)<0 when -3 < x < 4.