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

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Answer 1 · 2016-12-30T17:51:08

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

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 $x in ] -1/16 , 1 [$

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