Question

How do you solve `x^4-5x^2<=-4`?

Answer 1

The answer is `x in [-2, -1 ] uu [1, 2 ]`

Explanation:

We need

`a^2-b^2=(a+b)(a-b)`

Let rewrite the inequality

`x^4-5x^2+4<=0`

Let's factorise

`(x^2-1)(x^2-4)<=0`

`(x+1)(x-1)(x+2)(x-2)<=0`

Let `f(x)=(x+1)(x-1)(x+2)(x-2)`

Let's build a sign chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaa)` `-2` `color(white)(aaaa)` `-1` `color(white)(aaaa)` `+1` `color(white)(aaaa)` `+2` `color(white)(aaaa)` `+oo`

`color(white)(aaaa)` `x+2` `color(white)(aaaaaa)` `-` `color(white)(aaaa)` `+` `color(white)(aaaa)` `+` `color(white)(aaaa)` `+` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `x+1` `color(white)(aaaaaa)` `-` `color(white)(aaaa)` `-` `color(white)(aaaa)` `+` `color(white)(aaaa)` `+` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `x-1` `color(white)(aaaaaa)` `-` `color(white)(aaaa)` `-` `color(white)(aaaa)` `-` `color(white)(aaaa)` `+` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `x-2` `color(white)(aaaaaa)` `-` `color(white)(aaaa)` `-` `color(white)(aaaa)` `-` `color(white)(aaaa)` `-` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `f(x)` `color(white)(aaaaaaa)` `+` `color(white)(aaaa)` `-` `color(white)(aaaa)` `+` `color(white)(aaaa)` `-` `color(white)(aaaa)` `+`

Therefore,

`f(x)<=0` when `x in [-2, -1 ] uu [1, 2 ]`

Answer 2

By graphical method, `x in [-1, -2] and x in [1. 2]`

Explanation:

graph{y-x^4+5x^2-4>=0 [-3.03, 3.048, -1.528, 1.51]}

Let y-(x^4-5x^2+4)>=0.

The section of the graph by the x-axis ( y = 0 ) depicts the solution

-`(x^2-5x+4)>=0`, giving

`x^2-5x+4<=0`