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If −42𝑥< −41, then ＿.
If negative forty-two times 𝑥 is less than negative forty-one, then blank.
Here we have our inequality: negative forty-two has been multiplied to 𝑥. That means, to get rid of it, we need to divide both sides by negative forty-two.
On the left, the negative forty-twos cancel. On the right, the negatives turn positive because a negative divided by a negative makes a positive. And then forty-one forty-seconds doesn’t reduce. Now it’s important to know that we divided by a negative.
This means our less than sign needs to be turned into a greater than sign. So 𝑥 is greater than forty-one forty-seconds.
Now while this is true, let’s try to make sense of it. Why did that sign have to change directions?
This means if we’re gonna take negative forty-two and multiply it by a number, in order for that to stay less than negative forty-one, we have to plug in numbers that are greater than forty-one forty-seconds. So let’s go ahead and actually try forty-one forty-seconds. If we plug that in, that’s not greater than forty-one forty-seconds, so it shouldn’t work. Let’s try.
So we have negative forty-two times forty-one forty-seconds, which you could put negative forty-two over one. And we can see that the forty-twos cancel, so we have a negative times forty-one.
So we have negative forty-one in the numerator. Then on the denominator, it’s just a one times a one, so we have negative forty-one is less than negative forty-one, and that’s not true.
Negative forty-one is equal to negative forty-one, so we need the left-hand side to stay less than negative forty-one. Now on a number line, in order to be less than negative forty-one, it needs to be more negative. So it has to go to the left.
So this is why we have to plug in numbers that are greater than forty-one forty-seconds because this will keep our numbers more negative, larger in the negative way. So again if negative forty-two 𝑥 is less than negative forty-one, then 𝑥 must be greater than forty-one forty-seconds.
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