Question

How do solve `x^2<10-3x` and write the answer as a inequality and interval notation?

Answer 1

The answer is `-5 < x <2`
`x in ]-5, 2[`

Explanation:

Let's rewrite the inequality

`x^2+3x-10<0`

Let's factorise

`(x-2)(x+5)<0`

Let `f(x)=(x-2)(x+5)`

We can build the sign chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaa)` `-5` `color(white)(aaaa)` `2` `color(white)(aaaaa)` `+oo`

`color(white)(aaaa)` `x+5` `color(white)(aaaaa)` `-` `color(white)(aaaa)` `+` `color(white)(aaaa)` `+`

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

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

Therefore,

`f(x)<0` when `x in ]-5, 2[`

or `-5 < x <2`