Question

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

if possible with the explanation as well ​

Answer 1

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 = (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 - ( - (1)/(5) )| < c`

`|x + (1)/(5) | < c`

Let use roots 3/5

`| (3)/(5) + (1)/(5) | = (4)/(5)`

-1 works as well.

`| - 1 + (1)/(5) | = | - (4)/(5) | = (4)/(5)`

So the absolute value equation is

`|x + (1)/(5) | < (4)/(5)`