Question 693728: Solve the inequality. Y/-4<-3
Answer by RedemptiveMath(80)   (Show Source):
You can put this solution on YOUR website! You would solve this inequality just like you would if it were an equation. We need to solve for the variable by getting the variable by itself on one side. However, there are two key details right now that you need to keep in mind: The first is that we are dealing with inequality symbols besides equal signs, and the second is that we need to be mindful of negatives in inequalities. Let us see why these two things are important. Let us make this an equation to explain what I'm saying. So,
Y/-4 = -3.
If you were to solve this, you would find that y = 12 by multiplying both sides by -4. However, this is what happens when we multiply (or divide) both sides of an inequality by a negative number:
Y/-4 < -3
-4(Y/-4) < -4(-3)
Y > 12.
As you can see, with inequalities the sign changes directions when we multiply or divide by a negative number to both sides of the inequality. This can be explained in a simple manner as so:
If this was an equation and you found y, you'd know that y equals 12 by rearranging the equation around using multiplication. You would also be confident that you had the right answer when you plug y = 12 back into the equation. 12/-4 does equal -3, so you know that you are correct. However, in inequalities we are dealing with a range of numbers, not just one number as in the simple equation above. If you solve the inequality as an equation and say y < 12, you might feel confident that you have the right answer. But you would find that if you plugged in any number that is less than 12 (hence the range of numbers) back into the inequality, you'd find some contradiction. Using 11, we would find the inequality 11/-4 < -3, but this isn't true. -2.75 should be greater than -3, not the other way around. So the simple explanation is we switch the sign around when we multiply or divide by a negative in an inequality. To check yourself to provide extra confidence, just plug any number in the range you solved for back into the inequality and see if you're right or wrong. If you're wrong and you are dealing with multiplication or division of negatives, it is most likely because you didn't switch the sign's direction.
So, to check ourselves on this problem, if the answer is y > 12, then 13 should work:
Y/-4 < -3
13/-4 < -3 (is 13/-4 less than -3?)
-3.25 < -3 (13/-4 or -3.25 is less than -3).
Therefore, y must be greater than 12.
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