x, y, and z are coordinates of three points on the number line above. COLUMN A: xy COLUMN B: xz
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When comparing two products that share a common positive factor, you can simply compare the other factors. Since \(x\) is positive and common to both columns, comparing \(xy\) and \(xz\) is the same as comparing \(y\) and \(z\).
The relationship cannot be determined from the information given.
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The Correct Option isB
Solution and Explanation
Step 1: Understanding the Concept:
We need to compare the products \(xy\) and \(xz\) based on the positions of \(x, y,\) and \(z\) on the number line. Step 2: Detailed Explanation:
From the number line, we can determine the signs and relative magnitudes of the variables:
\(x\) is between 0 and 1, so \(x\) is a positive fraction (\(0<x<1\)).
\(y\) is between \(x\) and \(z\), and is greater than 1. So \(y\) is a positive number greater than 1.
\(z\) is to the right of \(y\), so \(z\) is a positive number greater than \(y\). Thus, \(z>y>1\).
Column A: \(xy\)
Column B: \(xz\)
We are comparing \(xy\) and \(xz\). Since \(x\) is a positive number (\(x>0\)), we can divide both sides of the inequality \(y<z\) by \(x\) without changing the direction of the inequality sign. Or, more simply, we can multiply both sides of \(y<z\) by the positive number \(x\).
Given \(y<z\), and \(x>0\), it follows that:
\[ x \cdot y<x \cdot z \]
\[ xy<xz \]
Therefore, the quantity in Column B is greater than the quantity in Column A.
Step 3: Final Answer:
Since \(x\) is positive and \(z\) is greater than \(y\), the product \(xz\) must be greater than the product \(xy\).
