Question

analyze determine whether the following statement is sometimes, always, or never true. justify your argument. \xy is on a number line and point w is \\(\frac{2}{5}\\) of the distance from x to y, then the coordinate of point w is greater than the coordinate of point x select choice \\(\boldsymbol{\downarrow}\\), if the coordinate of x is 0 and the coordinate of y is negative, then the coordinate of w will be select choice \\(\boldsymbol{\downarrow}\\) and select choice \\(\boldsymbol{\downarrow}\\) the coordinate of x. if the coordinate of x is positive and the coordinate of y is greater than the coordinate of x, then the coordinate of w will be select choice \\(\boldsymbol{\downarrow}\\) than the coordinate of x.
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Explanation:

Step1: Define Variables and Distance

Let the coordinate of \( X \) be \( x \) and the coordinate of \( Y \) be \( y \). The distance between \( X \) and \( Y \) is \( |y - x| \). Point \( W \) is \( \frac{2}{5} \) of this distance from \( X \), so the coordinate of \( W \), \( w \), is \( x+\frac{2}{5}(y - x)=\frac{3}{5}x+\frac{2}{5}y \) (assuming \( W \) is between \( X \) and \( Y \); we'll check direction later).

Step2: Case 1 - \( x = 0 \), \( y<0 \)

Substitute \( x = 0 \) into \( w \): \( w=\frac{2}{5}y \). Since \( y<0 \), \( \frac{2}{5}y<0 \), and \( x = 0 \), so \( w

Step3: Case 2 - \( x>0 \), \( y>x \)

Substitute into \( w=\frac{3}{5}x+\frac{2}{5}y \). Let's compare \( w \) and \( x \): \( w - x=\frac{2}{5}(y - x) \). Since \( y>x \), \( y - x>0 \), so \( w - x>0 \), meaning \( w>x \). Thus, the coordinate of \( W \) is greater than that of \( X \).

Step4: Determine "Sometimes, Always, Never"

For the first part ( \( x = 0 \), \( y<0 \) ), \( W \) is never greater (always less). For the second part ( \( x>0 \), \( y>x \) ), \( W \) is always greater. But the original statement's first part: "if the coordinate of \( X \) is 0 and the coordinate of \( Y \) is negative, then the coordinate of \( W \) will be [greater/less] than \( X \)" – from Step2, it's less. The second part: "if the coordinate of \( X \) is positive and \( Y \) is greater than \( X \), then \( W \) will be [greater/less] than \( X \)" – from Step3, it's greater. But the initial question is about whether the statement "If \( \overline{XY} \) is on a number line and point \( W \) is \( \frac{2}{5} \) of the distance from \( X \) to \( Y \), then the coordinate of point \( W \) is greater than the coordinate of point \( X \)" is sometimes, always, or never true.

From Step2, when \( x = 0 \), \( y<0 \), \( w0 \), \( y>x \), \( w>x \). So it's sometimes true (true in some cases, false in others).

Answer:

The statement is sometimes true. For the first "Select Choice" (when \( X = 0 \), \( Y<0 \)): the coordinate of \( W \) is less than that of \( X \). For the second "Select Choice" (when \( X>0 \), \( Y>X \)): the coordinate of \( W \) is greater than that of \( X \).