Question

which of the following is a solution to the inequality below?
$42 \leq \frac{132}{x}$
$x = 11$ $x = 2$ $x = 6$ $x = 4$

Explanation:

Step1: Analyze the inequality \(42 \leq \frac{132}{x}\)

We need to check for each given value of \(x\) (11, 2, 6, 4) whether the inequality holds.

Step2: Check \(x = 11\)

Substitute \(x = 11\) into \(\frac{132}{x}\), we get \(\frac{132}{11}=12\). Now check if \(42\leq12\). Since \(42>12\), this is false.

Step3: Check \(x = 2\)

Substitute \(x = 2\) into \(\frac{132}{x}\), we get \(\frac{132}{2} = 66\). Now check if \(42\leq66\). Since \(42<66\), this is true? Wait, no, wait. Wait, the inequality is \(42\leq\frac{132}{x}\), so when \(x = 2\), \(\frac{132}{2}=66\), and \(42\leq66\) is true? Wait, but let's check other values too. Wait, maybe I made a mistake. Wait, let's check \(x = 2\): \(\frac{132}{2}=66\), \(42\leq66\) is true. Wait, but let's check \(x = 4\): \(\frac{132}{4}=33\), \(42\leq33\) is false. \(x = 6\): \(\frac{132}{6}=22\), \(42\leq22\) is false. \(x = 2\): \(\frac{132}{2}=66\), \(42\leq66\) is true. Wait, but wait, maybe I messed up the direction. Wait, the inequality is \(42\leq\frac{132}{x}\), so we can also solve the inequality:
Multiply both sides by \(x\) (we need to consider the sign of \(x\). If \(x>0\), the inequality direction remains the same; if \(x < 0\), it reverses. But since \(x\) is in the denominator of \(\frac{132}{x}\) and we have positive values for \(x\) (11, 2, 6, 4 are positive), so we can multiply both sides by \(x\) (positive) to get \(42x\leq132\), then \(x\leq\frac{132}{42}=\frac{22}{7}\approx3.14\). So \(x\) must be less than or equal to approximately 3.14. Among the given options, \(x = 2\) is less than 3.14, \(x = 4\) is more than 3.14, \(x = 6\) more, \(x = 11\) more. So \(x = 2\) is the solution. Let's verify again: for \(x = 2\), \(\frac{132}{2}=66\), and \(42\leq66\) is true. For \(x = 4\), \(\frac{132}{4}=33\), \(42\leq33\) is false. For \(x = 6\), \(\frac{132}{6}=22\), \(42\leq22\) is false. For \(x = 11\), \(\frac{132}{11}=12\), \(42\leq12\) is false. So the only one that works is \(x = 2\).

Answer:

\(x = 2\) (i.e., the option "x = 2")