does adding a positive or negative (subtracting) number change the inequality symbol?

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answers:
la console say: No. When you make something on one side, you must make the similar thing to the other side.


2x - 3 < 4x - 5 → you add 3 both sides

2x - 3 + 3 < 4x - 5 + 3 → you simplify

2x < 4x - 2 → you subtract 4x both sides

2x - 4x < 4x - 2 - 4x → you simplify

- 2x < - 2 → you divide both sides by (- 1), as (- 1) is negative, you CHANGE the direction

- 2x * (- 1) > - 2 * (- 1) → you simplify

2x > 2 → you divide both sides by (2), as (2) is positive, you DON'T CHANGE the direction

2x/2 > 2/2 → you simplify

x > 1
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Dixon say: No. The only time it changes the symbol is if you negate both sides, ie multiply or divide the whole of both sides by a negative number. The negative reverses the direction of the values, so inevitably the inequality need to be in the opposite direction too.
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Jeffrey K say: No. Only multiplying or dividing by a negative number reverses the inequality sign.
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Puzzling say: No. You can add or subtract the same value (positive or negative) on both sides and it DOESN'T affect the direction of the inequality.

However, if you *multiply* or *divide* by a negative number, that DOES change the direction of the inequality.

For example,

We know that:
1 < 7

But if we multiply both sides by -1, we should get:
-1 > -7

(It's the same thing with variables and expressions, but it should be even clearer with actual numbers).
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Steve A say: No. Multiplying or dividing by a negative changes the inequality.

x + 3 > 4
x + 3 (-3) > 4 (-3)
x > 1

-x + 3 > 4
-x > 1
x < -1
You can get the same answer, in a different order, by adding x to each side instead of multiplying by -1.
-x > 1
-x (+x) > 1 (+x)
0 > 1 + x
0 (-1) > 1 (-1) +x
-1 > x (Same answer as with multiplying by -1 with the order reversed.)
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