Question

How do you solve `abs(4x-5)>16`?

Answer 1

`abs(4x-5) > 16` when `x<-11/4` and `x>21/4`

Explanation:

The "absolute value" function is defined as :

When `x in RR,`

`|x|=x,ifx≥0`
`|x|=−x,ifx<0`

And `4x-5>=0` when `x>=5/4`

Then, `abs(4x-5)=4x-5, if x>=5/4`
And, `abs(4x-5)=-(4x-5)=-4x+5, if x<5/4`

To resolve the inequality, we have to separate them in two parts :
when `x>=5/4` and when `x<5/4`

If `x>=5/4` :
`abs(4x-5) > 16 <=> 4x-5 > 16 <=> 4x > 21 <=> color(red)(x>21/4)`


If `x<5/4` :
`abs(4x-5) > 16 <=> -4x+5 > 16 <=> -4x > 11 <=> -x>11/4`

Recall : when we multiply or divide each side of an inequality by a negative number, we have to inverse the sign of the inequality

Ex : If `-a>0` then `acolor(red)<0` (multiplied by `-1`)


Then `abs(4x-5) > 16 <=> color(red)(x<-11/4)`


Therefore, `abs(4x-5) > 16` when `x<-11/4` and `x>21/4`


Other notation :
`abs(4x-5) > 16` if `x in ]-oo;-11/4[uu]21/4;+oo[`