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When graphing inequalities on a number line, if you multiply by a negative, you have to change the sign to the opposite sign. For example, if an inequality is x/-4>4, and you multiply both sides by -4, the inequality becomes x<-1. However, if the inequality is x/4>-4, and you multiply both sides by 4, the inequality is x>-1, without using the opposite sign... but the communitive property of multiplication says -4 times 4 is the same as 4 times -4. Why does this not apply to inequalities?
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Say we have 3 < 4. If you multiply both sides by some negative number, like -1, then you get -4 and -3. You could visualize this as flipping the number line around 180 degrees. 3 is further to the left than 4, so we can say that 3 < 4. However, -4 is further to the left than -3, so -4 < -3. The general logic of flipping the number line (or reflecting through the origin) applies generally, which means you have to invert the inequality when multiplying by or dividing by negative values.
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The commutive property of multiplication applies to expressions-- not to equations or inequalities.

-2x>6 can be solved without flipping the inequality sign.

Add 2x to both sides:

0>2x+6

Subtract 6 from both sides:

-6>2x

Divide both sides by 2:

-3>x

**x<-3**
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Thank you for helping me understand
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