Question

How do you solve `16x^2<15x+1`?

Answer 1

The answer is `x in ] -1/16 , 1 [`

Explanation:

Let's rewrite the equation

`16x^2-15x-1<0`

Let `f(x)=16x^2-15x-1`

The domain of `f(x)` is `D_f(x)=RR`

We can factorise the RHS

`16x^2-15x-1=(16x+1)(x-1)`

Now we can establish the sign chart

`color(white)(aaaa)` `x` `color(white)(aaaaaaaa)` `-oo` `color(white)(aaaaa)` `-1/16` `color(white)(aaaaa)` `1` `color(white)(aaaaa)` `+oo`

`color(white)(aaaa)` `16x+1` `color(white)(aaaaaaaa)` `-` `color(white)(aaaaaaa)` `+` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `x-1` `color(white)(aaaaaaaaaa)` `-` `color(white)(aaaaaaa)` `-` `color(white)(aaaa)` `+`

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

Therefore,

`f(x)<0`, when `x in ] -1/16 , 1 [`