Question

How do you solve `abs(6 - 5y )>2`?

Answer 1

The solution is `y in ( -oo, 4/5)uu(8/5,+oo)`

Explanation:

There are `2` solutions for absolute values.

`|6-5y|>2`

`6-5y>2` and `-6+5y>2`

`5y<4` and `5y>8`

`y<4/5` and `y>8/5`

The solutions are

`S_1=y in( -oo, 4/5)`

`S_2=y in (8/5,+oo)`

Therefore,

`S=S_1uuS_2`

`=y in ( -oo, 4/5)uu(8/5,+oo)`

graph{(y-|6-5x|)(y-2)=0 [-5.55, 6.934, -0.245, 5.995]}

Answer 2

`(-oo,4/5)uu(8/5,+oo)`

Explanation:

`"inequalities of the form "|x|>a`

`"always have solutions of the form"`

`x< -a" or "x > a`

`"thus we have to solve 2 inequalities"`

`6-5y < -2" or "6-5y>2`

`color(blue)"first solution"`

`6-5y < -2larr"subtract 6 from both sides"`

`rArr-5y < -8larr"divide both sides by "-5`

`rArry>8/5larrcolor(blue)"reverse direction of sign"`

`color(blue)"second solution"`

`6-5y>2larr"subtract 6 from both sides"`

`rArr-5y> -4larr"divide both sides by " -5`

`y< 4/5larrcolor(blue)"reverse direction of sign"`

`"combining the solutions gives"`

`(-oo,4/5)uu(8/5,+oo)`