Question

When you are solving inequalities, when does the sign change and when does it not?

For example, in -4x -5 > 8
I know I add 5 to both sides, then I have -4x > 13
Divide both sides by -4 and it becomes x < -3.25

But WHY does the direction of the sign change?

In 9x + 6 > 7, I subtract 6 from both sides to get 9x > 1
Divide both sides by 9 and I get x > 1/9, so the sign stays the same.

Could someone please explain for me?

Answer 1

Whenever you multiply or divide across an inequality by a negative number, you change the direction.

Think about 2 > -4

Multiply both sides by a -3

-6 > 12 NO... change the inequality -6<12

Answer 2

the "direction change" rule is just a shortcut for when things move from one side of the inequality to the other. When adding and subtracting, or multiplying/dividing by a positive value, the direction never changes.

-4x < 8
divide by -4 and change direction:
x > -2

OR

-4x < 8
0 < 8 + 4x
-8 < 4x
-2 < x

they both end up the same