Question

Show that 5x^2 + 2x - 3 < 0 can be written in the form | x + 1/5 | < 4/5

Answer 1

Step-by-step explanation:

First let solve the inequality

$5 {x}^{2} + 2x - 3 < 0$

Factor by grouping

$5 {x}^{2} + 5x - 3x - 3 < 0$

$5x(x + 1) - 3(x + 1)$

So the factor are

$(5x - 3)(x + 1)$

So the factor are

$x = \frac{3}{5}$

and

$x = - 1$

Solutions to a quadratic can be represented by a absolute value equation because remeber quadratics

creates 2 roots and/or double roots.

The inequality

$|x - b| < c$

works as

b is the midpoint between 2 roots. And c is the

$|x + b| = c$

We know that the midpoint between both roots is-1/5.

so

$|x - ( - \frac{1}{5} )| < c$

$|x + \frac{1}{5} | < c$

Let use roots 3/5

$| \frac{3}{5} + \frac{1}{5} | = \frac{4}{5}$

-1 works as well.

$| - 1 + \frac{1}{5} | = | - \frac{4}{5} | = \frac{4}{5}$

So the absolute value equation is

$|x + \frac{1}{5} | < \frac{4}{5}$