Question

in order to qualify for a role in a play, an actor must be taller than 64 inches but shorter than 68 inches. the inequality $64 < x < 68$, where $x$ represents height, can be used to represent the height range. which is another way of writing the inequality?

$x < 64$ or $x < 68$

$x < 64$ and $x < 68$

$x > 64$ or $x < 68$

$x > 64$ and $x < 68$

Explanation:

Step1: Analyze the original inequality

The original inequality is \(64 < x < 68\). This means that \(x\) has to satisfy two conditions at the same time: \(x\) is greater than 64 (\(x>64\)) and \(x\) is less than 68 (\(x < 68\)). The word "and" is used here because \(x\) must meet both conditions simultaneously. If we used "or", it would mean that \(x\) only needs to satisfy one of the conditions, which is not the case here. For example, if we took \(x>64\) or \(x < 68\), a value like \(x = 60\) (which is less than 64) would satisfy the "or" statement but does not satisfy the original inequality \(64 < x < 68\). Similarly, a value like \(x=70\) (which is greater than 68) would satisfy the "or" statement but not the original inequality. So we need the "and" of the two inequalities \(x > 64\) and \(x < 68\).

Step2: Evaluate each option

• Option 1: \(x < 64\) or \(x < 68\): This is incorrect because it allows values less than 64, which do not satisfy the original inequality.
• Option 2: \(x < 64\) and \(x < 68\): This is incorrect because it only considers values less than 64, which do not satisfy the original inequality.
• Option 3: \(x>64\) or \(x < 68\): This is incorrect because it allows values greater than 68 or less than 64, which do not satisfy the original inequality.
• Option 4: \(x>64\) and \(x < 68\): This is correct as it represents the two conditions that \(x\) must be greater than 64 and less than 68 at the same time, which is exactly what the original inequality \(64 < x < 68\) means.

Answer:

\(x>64\) and \(x < 68\) (the fourth option)