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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 ​

User Alephnerd
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1 Answer

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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)

User Win
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