Question

How do you solve `|7 - 6x| <= -4`?

Answer 1

`x >= +11/6 or x <=1/2`

Set the +value `<= -4 and -"value" <= -4` and solve.

Explanation:

The absolute value can be either a negative or a positive value. It is the distance from the value to zero. So the answer that comes out is the same regardless if the answer inside is negative or positive.

Set `+1 xx ( 7 - 6x) <= -4` and solve for the positive value.

This gives

`7- 6x <= -4 " "` subtract -7 from both sides.

`7-7 - 6x <= -4 -7" "` This gives

`-6x <= -11 " "` divide everything by -1 . The sign will change.

`(- x)/-1 >= (-11)/(6xx-1)" "` This gives

`x >= + 11/6`

`( -1)/-1 = +1` the opposite of -a - is +a (Dividing by a negative always gives you the opposite of what you start with)

When you divide an inequality by a negative number the inequality sign in the middle changes around.

`(-11)/(-1 xx6) = + 11/6` (Dividing by a negative always gives you the opposite sign of what you start with)

Then set `-1( 7-6x ) <= -4" "` and solve for x .This gives

`-7 + 6x <= -4" "` add seven to both sides

`-7 +7 + 6x <= -4 + 7 " "` this gives

`6x <= + 3 " "` Divide both sides by 6

`(6x)/6<= +3/6" "` this gives

`x <= 1/2`

Note that as the x is positive, dividing by +1 would not change any of the values, so is not necessary.