How do you solve x24x using a sign chart?

1 Answer
Nov 16, 2016

Move every term to one side, factor the expression, and determine when each factor is+or. Multiply these+andintervals from the factors to get the+andintervals for the product (and thus the original inequality).

Explanation:

x24x
x24x0 Move everything to one side
x(x4)0 Factor the expression

This is now written as the product of two factors, x and x4. The values of x that make the factors 0 are the "points of interest", because it is only at these points where the value of x24x could change from being0 to not being0. We set each factor equal to 0 and solve for x:

x=0orx4=0
x=0orx=4

So we have three intervals to examine: below 0, between 0 and 4, and above 4.

A sign chart can be made like this:
. . . . . . . . . .---------0----------4-----------
x
x4
. . . . . . . . . .----------------------------------
x(x4)

Fill in the sign chart with + and - signs to reflect where each factor is0. For example: the factor x4 is negative when x<4; it's 0 when x=4; and it's positive when x>4.

. . . . . . . . . .---------0---------4-----------
x . . . . . . . . 0+++++
x4. . . . . 0+++
. . . . . . . . . .----------------------------------
x(x4)

To complete the bottom row of the sign chart, each interval is filled with the product of all the signs above it. For example, when x<0, () x () = (+).

. . . . . . . . . .---------0--------4-----------
x . . . . . . . . 0+++++
x4 . . . . 0+++
. . . . . . . . . .----------------------------------
x(x4) . ++00+++

This tells you that x(x4) (in other words, x24x) will be negative only when x is between 0 and 4, and it will be 0 at those endpoints, i.e. x24x0 when 0x4. And since x24x0 is equivalent to our original inequality x24x, we are done.

The solution is 0x4.
Or x[0,4] if you prefer interval notation.

Go ahead and try some values for yourself: say x=3. We should expect the inequality to be true. Well, is it?

x24x
324(3) (when x=3)
912
Which is true!