Question

How do you solve `x^4-10x^2+9>=0` using a sign chart?

Answer 1

Solution is `-oo <= x <= -3` or `-1 <= x <= 1`, or `3 <= x <= oo` and in interval form it can be written as `[-oo,-3]uu[-1,1]uu[3,oo]`

Explanation:

Let us first factorize `x^4-10x^2+9`.

`x^4-10x^2+9=x^4-9x^2-x^2+9`

= `x^2(x^2-9)-1(x^2-9)= (x^2-9)(x^2-1))`

= `(x+3)(x-3)(x+1)(x-1)`

Hence we have to solve the inequality

`(x+3)(x-3)(x+1)(x-1)>=0`

From this we know that the product `(x+3)(x-3)(x+1)(x-1)` has to be zero or positive. It is apparent that sign of binomials `(x+3)`, `(x+1)`, `(x-1)` and `(x-3)` will change around the values `-3`. `-1`, `1` and `3` respectively. In sign chart we divide the real number line using these values, i.e. below `-3`, between `-3` and `-1`, between `-1` and `1`, between `1` and `3` and above `3` and see how the sign of `(x+3)(x-3)(x+1)(x-1)` changes.

Sign Chart

`color(white)(XXXXXXXX)-3color(white)(XXXX)-1color(white)(XXXX)1color(white)(XXXX)3`

`(x+3)color(white)(XX)-ive color(white)(XX)+ive color(white)(XXX)+ive color(white)(XX)+ive color(white)(XX)+ive`

`(x+1)color(white)(XX)-ive color(white)(XX)-ive color(white)(XXX)+ive color(white)(XX)+ive color(white)(XX)+ive`

`(x-1)color(white)(XX)-ive color(white)(XX)-ive color(white)(XXX)-ive color(white)(XX)+ive color(white)(XX)+ive`

`(x-3)color(white)(XX)-ive color(white)(XX)-ive color(white)(XXX)-ive color(white)(XX)-ive color(white)(XX)+ive`

`x^4-10x^2+9`
`color(white)(XXXXXX)+ive color(white)(Xx)-ive color(white)(XXX)+ive color(white)(XX)-ive color(white)(XX)+ive`

It is observed that `x^4-10x^2+9 >= 0`

when either `-oo <= x <= -3` or `-1 <= x <= 1`, or `3 <= x <= oo`, which is the solution for the inequality.

In interval notation, this can be written as `[-oo,-3]uu[-1,1]uu[3,oo]`

Answer 2

The values of x are
`-oo<=x<=-3` and `-1<=x<=1`

Explanation:

Let's start by factorising the expression

`x^4-10x^2+9=(x^2-1)(x^2-9)=(x+1)(x-1)(x+3)(x-3)`

So we can make the sign chart
`x` `color(white)(aaaaa)`-oo`color(white)(aaaa)`-3`color(white)(aaaa)` `-1` `color(white)(aaaa)` `1` `color(white)(aaaa)` `3` `color(white)(aaaa)` `+oo`
`x+3` `color(white)(aaa)` `-` `color(white)(aaa)`0`color(white)(aa)` `+` `color(white)(aaaa)` `+` `color(white)(aa)` `+` `color(white)(aa)` `+`
`x+1` `color(white)(aaa)` `-` `color(white)(aaa)` `color(white)(aaa)` `-` `color(white)(a)` `0` `color(white)(aa)` `+` `color(white)(aaa)` `+` `color(white)(aa)` `+`
You can continue the sign chart
and you wiil find that
`x^4-10x^2+9>=0` for `-oo<=x<=-3` and `-1<=x<=1`