Question

analyze determine whether the following statement is sometimes, a/ways, or never true. justify your argument. if $overline{xy}$ is on a number - line and point $w$ is $\frac{2}{3}$ of the distance from $x$ to $y$, then the coordinate of point $w$ is greater than the coordinate of point $x$.

Explanation:

Step1: Let the coordinate of $X$ be $x$ and the coordinate of $Y$ be $y$, and the coordinate of $W$ be $w$. The distance from $X$ to $Y$ is $d = |y - x|$. Given that $W$ is $\frac{2}{3}$ of the distance from $X$ to $Y$ from $X$, so $w=x+\frac{2}{3}(y - x)=\frac{1}{3}x+\frac{2}{3}y$ (assuming $y\geq x$; if $x > y$, then $w=x-\frac{2}{3}(x - y)=\frac{1}{3}x+\frac{2}{3}y$).

Step2: Analyze the statement. We want to know if $w>x$. Substitute $w=\frac{1}{3}x+\frac{2}{3}y$ into $w > x$. We get $\frac{1}{3}x+\frac{2}{3}y>x$. Rearranging gives $\frac{2}{3}y>\frac{2}{3}x$, or $y > x$.

Step3: Conclusion. If $y>x$, then the coordinate of $W$ is greater than the coordinate of $X$. But if $y = x$ (i.e., $X$ and $Y$ are the same - point), then $w=x$. So the statement "the coordinate of point $W$ is greater than the coordinate of point $X$" is sometimes true.

Answer:

The statement "the coordinate of point $W$ is greater than the coordinate of point $X$" is sometimes true. Justification: If $Y$ has a greater coordinate than $X$, then $W$ (which is $\frac{2}{3}$ of the distance from $X$ to $Y$ from $X$) has a greater coordinate than $X$. But if $X$ and $Y$ have the same coordinate, then $W$ has the same coordinate as $X$.