Video Transcript
Let 𝑐 and 𝑑 be numbers that satisfy the inequality 𝑐 plus 𝑑 is less than 𝑑 which is less than 𝑐 minus 𝑑. Which of the following must be true? I) 𝑐 is less than zero. II) 𝑑 is less than 𝑐. III) 𝑑 is less than zero. A) I, II, and III. B) I only. C) II and III. Or D) I and III only.
The first thing we can do is rearrange this inequality to see if we can find something that says 𝑐 is less than zero or 𝑑 is less than zero. We can do that by breaking this up into two different inequalities, 𝑐 plus 𝑑 is less than 𝑑 and 𝑑 is less than 𝑐 minus 𝑑. In our first inequality, we can subtract 𝑑 from both sides. The 𝑑s on the right cancel out, leaving us with zero. We’ve rearranged the first half of this inequality to say that 𝑐 is less than zero. And that means statement one must be true.
Now, that we’ve rearranged the first part so that it has 𝑐 is less than zero, we can take our second half and write it in terms of 𝑐 by adding 𝑑 to both sides. 𝑑 plus 𝑑 equals two 𝑑. And two 𝑑 has to be less than 𝑐. We can put these two statements back together, two 𝑑 is less than 𝑐 and 𝑐 is less than zero. And so, we have a new inequality that says two 𝑑 is less than 𝑐 which is less than zero.
We’ve already said that 𝑐 is less than zero. This means that 𝑐 is a negative. If multiplying two by some number has to be less than a negative number. That means that our 𝑑 value must also be negative. And that means statement three is also true. 𝑑 must be negative and 𝑐 must be negative.
Statement two is a little bit trickier to figure out. We know that two 𝑑 has to be less than 𝑐. But does that mean 𝑑 has to be less than 𝑐? To find out whether or not statement two is true, we’re going to look for a counterexample, an example that disproves the statement. A counterexample of statement two would be any place where 𝑑 is greater than 𝑐.
We already know two times 𝑑 has to be less than 𝑐 which has to be less than zero. We want to find numbers where 𝑑 is bigger than 𝑐 but still less than zero. For example, what if 𝑑 was equal to negative three? We’ll put a negative three dot on our number line. And then, we let 𝑐 be equal to negative four and put a dot there on our number line.
Is it true that two times negative three is less than negative four which is less than zero? Negative six is less than negative four which is less than zero. But in this case, our 𝑑 was greater than our 𝑐. Negative three is greater than negative four. This is a counterexample from 𝑑 is less than 𝑐. And while, sometimes, 𝑑 might be less than 𝑐, it is not always true. Our question is asking which must be true. Only one and three must be true. And that’s option D.
