Plugging In: DOZEN F for "Must Be" Questions

Given that $$a \lt 5$$ and $$b - a = 0$$, which of the following must be true?

Incorrect. [[Snippet]] If $$a = 4.5$$, then $$b = 4.5$$, not 5.

But you might say, $$b$$ cannot be equal to 5. You are right! That *doesn’t* mean that $$b \le 5$$ isn't true though. It is important to remember that **a variable can only have one singular value at a time**. When you see the inequality $$a\lt 5$$, it doesn't mean that $$a$$ is *every* number less than 5. Similarly, saying that $$b\le 5$$ doesn’t mean that $$b$$ is all of the numbers less than or equal to 5. This can be confusing because when you graph inequalities, you graph all of the *possible* solutions by shading part of the number line. The variable still only has a single value, and that value can be found anywhere within that shaded region. If you think about a number line, $$a\lt 5$$ means that $$a$$ can be found somewhere to the left of 5. Since $$a = b$$, it is then definitely true that $$b \le 5$$ because $$b\le 5$$ just means that $$b$$ can be found to the left of (or at) 5. If $$a$$ is to the left of 5, it is true that $$b$$ is to the left of 5 as well You could also think about a similar problem where the variable represents some value. Let $$x$$ be John's age. Suppose you were told that John's age is less than 30 years old ($$x \lt 30$$). Once you know that, you know that John's age is less than or equal to 40 years old ($$x \le 40$$). It doesn't matter that you know John can't be between the ages of 30 and 40. The question only asks which of the inequalities must be true.

Incorrect. [[Snippet]] If $$a = 4.5$$, then b = $$4.5$$, which is not greater than or equal to 5.

Incorrect. [[Snippet]] If $$a = 4.5$$, then $$b = 4.5$$, which is not greater than 5.

Incorrect. [[Snippet]] If $$a = 4.5$$, then $$b = 4.5$$, which is not less than 4.

Correct. [[snippet]] If you __Plug In__ $$a = 4.5$$, then $$b=4.5$$. That eliminates all answer choices except B. Thus, the correct answer is B.

$$b = 5$$

$$b \le 5$$

$$b \ge 5$$

$$b \lt 4$$

$$b \gt 5$$

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