Video: Identifying Equivalent Inequalities

Which of the following inequalities is equivalent to −4𝑥 ≤ −1? [A] 𝑥 = 1/4. [B] 𝑥 ≥ 1/4. [C] 𝑥 ≥ 4. [D] 𝑥 ≤ 1/4. [E] 𝑥 < 1/4.

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Video Transcript

Which of the following inequalities is equivalent to negative four 𝑥 is less than or equal to negative one? A) 𝑥 equals one-fourth. B) 𝑥 is greater than or equal to one-fourth. C) 𝑥 is greater than or equal to four. D) 𝑥 is less than or equal to one-fourth. Or E) 𝑥 is less than one-fourth.

Our inequality is negative four 𝑥 is less than or equal to negative one. And we want to rearrange this equation so only 𝑥 is on the left side. Right now, 𝑥 is being multiplied by negative four. To get 𝑥 by itself, we’ll need to do the opposite. We’ll need to divide by negative four. But if we do that on the left, we know we need to do that on the right.

And all of a sudden, we should be thinking we’re dividing by a negative and we’re working with inequalities. And that means, in order for this inequality to remain true, we need to flip the sign. Negative four 𝑥 divided by negative four equals 𝑥. And negative one divided by negative four equals one-fourth. And that means an equivalent inequality would be 𝑥 is greater than or equal to one-fourth, which is option B.

We can check and see if this is true. We’ve just said that 𝑥 can be anything that’s greater than or equal to one-fourth. We know that one is greater than or equal to one-fourth. So if we take our original inequality, negative four 𝑥 is less than or equal to negative one, and we plug in one, we should get a true statement. Is negative four less than negative one? Yes, that’s true, which means that one is a correct solution for 𝑥.

When we’re working with inequalities, I recommend checking your solutions, because what if you forgot to flip the sign? If you forgot, you would get 𝑥 is less than or equal to one-fourth. If you did that, you could’ve chosen something like negative one to check your solution. Negative one is less than one-fourth. But when you plug that in, negative four times negative one equals positive four. And positive four is not less than negative one, which would’ve told you that you’ve done something wrong. And you need to go back and check. In this case, 𝑥 must be greater than or equal to one-fourth, which is option B.

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