Video Transcript
If four 𝑥 plus seven is less than or equal to negative one, then five 𝑥 is less than or equal to blank.
Beginning with the given statement four 𝑥 plus seven is less than or equal to negative one, we want to rearrange it so that we can find what five 𝑥 is less than or equal to. We remember our rules for working with inequalities. If we’re adding and subtracting, an inequality still holds true when adding or subtracting the same amount from both sides. When it comes to multiplication and division, we have two cases. An inequality still holds true when both sides are multiplied or divided by the same positive number. When both sides are being multiplied or divided by a negative number, then the inequality symbol must change orientation for the statement to remain true.
We usually say that when they’re multiplied or divided by negative, you need to flip the sign. Using these tools, we can try to rearrange this equation and find out what five 𝑥 would be less than or equal to. First, we can subtract seven from both sides, which gives us four 𝑥 is less than or equal to negative eight. Now, we might think, “Okay, well, we could add 𝑥 to both sides,” which is true. That would be five 𝑥 on the left. But on the right, you would have is less than or equal to negative eight plus 𝑥. And that’s not so helpful, since we would prefer to have all the 𝑥’s on one side of the equation. And if you subtract 𝑥 from both sides, then we’re back where we started. Four 𝑥 is less than or equal to negative eight.
Another option we have is to find out how much one 𝑥 is less than or equal to. We get to that by dividing four 𝑥 by four and negative eight by four. Remember, we’re dividing by a positive number, so we do not change the sign. We get 𝑥 is less than or equal to negative two. But again, we want five 𝑥. We know that five 𝑥 is five times one 𝑥. And that means we could multiply both sides of the equation by five. Five times 𝑥 we can write as five 𝑥 and negative two times five is negative 10.
Now, we’ve multiplied by positive five on both sides which means we do not change the sign. Just because we have a negative number does not mean we change the sign. We always look at the operation we’re completing. And this operation was positive. So we have five 𝑥 is less than or equal to negative 10. And that means the missing blank for us will be negative 10.
