Question

the compound inequality $8.00 \leq x < 9.50$ represents all values, $x$, for which college students are paid hourly as teacher assistants. what is another way of writing this compound inequality?
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$$\begin{array}{|c|c|c|} \\hline x \\geq 8.00 \\text{ and } \\\\ x < 9.50 & x \\leq 8.00 \\text{ and } \\\\ x < 9.50 & x \\geq 8.00 \\text{ or } x < 9.50 \\\\ \\hline x \\leq 8.00 \\text{ or } x < 9.50 & & \\\\ \\hline\\end{array}$$

Explanation:

Step1: Understand compound inequality meaning

The compound inequality \(8.00 \leq x < 9.50\) means \(x\) is greater than or equal to \(8.00\) and \(x\) is less than \(9.50\). This is because the value of \(x\) has to satisfy both conditions simultaneously (it has to be at least \(8.00\) and also less than \(9.50\)).

Step2: Analyze each option

• First option: \(x \geq 8.00\) and \(x < 9.50\) – this matches the meaning of \(8.00 \leq x < 9.50\).
• Second option: \(x \leq 8.00\) and \(x < 9.50\) – this would mean \(x\) is at most \(8.00\) and less than \(9.50\), which is not the same as the original inequality (original requires \(x\) to be at least \(8.00\)).
• Third option: \(x \geq 8.00\) or \(x < 9.50\) – "or" means \(x\) can satisfy either condition, but the original inequality requires \(x\) to satisfy both (so "and" is needed, not "or").
• Fourth option: \(x \leq 8.00\) or \(x < 9.50\) – this is also incorrect as it uses "or" and the wrong lower bound.

Answer:

\(x \geq 8.00\) and \(x < 9.50\) (the first option)