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# 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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