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?

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

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

When solving inequalities, whether the direction of the sign changes or stays the same depends on two factors: the operation performed and the positioning of the variable in the inequality.

1. Addition/Subtraction:
When you add or subtract a value to both sides of an inequality, the direction of the sign remains the same.
For example, in -4x - 5 > 8:
-4x - 5 + 5 > 8 + 5
-4x > 13
Dividing both sides by -4, you get:
(-4x) / -4 < 13 / -4
x < -3.25
In this case, the sign (>) did not change, as we added the same value (5) to both sides.

2. Multiplication/Division:
When you multiply or divide both sides of an inequality by a negative value, the direction of the sign changes.
For example, in -4x > 13:
Dividing both sides by -4, you get:
-4x / -4 < 13 / -4
x > -3.25
In this case, the sign (<) changed because we divided both sides by a negative value. Dividing by a negative value flips the inequality sign.

3. Variable Positioning:
The positioning of the variable also affects whether the sign changes or stays the same. When you move the variable from one side of the inequality to the other, the direction of the sign changes.
For example, in 9x + 6 > 7:
Subtracting 6 from both sides, you get:
9x + 6 - 6 > 7 - 6
9x > 1
Dividing both sides by 9, you get:
(9x) / 9 > 1 / 9
x > 1/9
In this case, the sign (> ) stayed the same because the variable remained on the same side of the inequality.

In summary, when solving inequalities, the direction of the sign stays the same when you add or subtract on both sides and when you keep the variable on the same side. However, the direction of the sign changes when you multiply or divide both sides by a negative value or when you move the variable across the inequality.