Question

How do you solve `x^2<=4x` using a sign chart?

Answer 1

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

Explanation:

`x^2<=4x`
`x^2-4x<=0` Move everything to one side
`x(x-4)<=0` Factor the expression

This is now written as the product of two factors, `x` and `x-4`. 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 `x^2-4x` could change from being`<=0` to not being`<=0`. We set each factor equal to `0` and solve for `x`:

`x=0 or x-4=0`
`=>x=0 or x=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`
`x-4`
. . . . . . . . . .----------------------------------
`x(x-4)`

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

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

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

This tells you that `x(x-4)` (in other words, `x^2-4x`) will be negative only when `x` is between 0 and 4, and it will be 0 at those endpoints, i.e. `x^2-4x<=0` when `0<=x<=4`. And since `x^2-4x<=0` is equivalent to our original inequality `x^2<=4x`, we are done.

The solution is `0<=x<=4`.
Or `x in [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?

`x^2<=4x`
`<=>3^2<=4(3)` (when `x=3`)
`<=>9<=12`
Which is true!