dividing inequalities by negative number
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Answers:Consider the statement 4 < 7, which is true. If you multiply both sides by -1 but you did not change the direction of the sign you would have -4 < -7, which is a false statement. Therefore, the direction must be changed, as well: -4 > -7. Yes, this holds true for equations, as well. 2 - 3x > 5 -3x > 3 (and now divide by -3) x < -1
Answers:No need to "memorize" this. I always suggest people prove it for themselves. First, wriite down an inequality that you know is true, and do whatever you want to it to see how you would have to change the inequality sign for each test. For instance, 2 is less than 3, so in terms of an inequality we can write 2 < 3 What happens if you multiply both sides by a negative number, say -5? 2*(-5) < 3*(-5) -10 < -15 but that is not true, so you should write: -10 > -15, see how the sign flipped? This is not so much a question of why, as just learning a set of rules. What if you divide the statement 2 < 3, by 5? 2/5 < 3/5 ? Sure, that is also true. What about dividing by -5? -2/5 < -3/5? No, -3/5 is more negative than -2/5, so we would have to write -2/5 > -3/5 for the inequalilty to be true. So, we conclude that multiplying or dividing by a negative number forces a flip of the inequality sign to have consistent math. (B) for equations, there is no inequality sign to flip, an equation is an 'equality' (i.e. has an equal sign, so this cannot happen for equations). Asa a final note, inequalities do not need to just be wtih numbers, they can involve variables like x. Make sense?
Answers:Think about it like this: the larger the number is when positive, the smaller the number is when negative because it's far away from zero, but far in a negative direction is smaller. Also, think about it like this: -2x + 8 > 0 -2x > -8 x ? 4 What if instead, you add 2x to both sides instead of dividing by -2 at the end? -2x + 8 > 0 8 > 2x 4 > x
Answers:Simple illustration: Clearly 5 >1 but -5 < -1 Again, 2 > -3, but -2 < 3 Another way of explaining this is that for two positive numbers, the one further from 0 is greater than (i.e. to the right of) the other, but for two negative numbers the one further from 0 is less than (i.e. to the left of) the other. More abstract answer: If x > y, then for positive p, px > py Add -px - py to both sides: -py > - px which is equivalent to -px < -py As for equations, well ... have you thought about this at all for yourself, or do you want someone to supply you with a completed homework answer without any input from you? What happens if you reverse "="? It's still "=", isn't it. You can figure an example.