Question

analyze determine whether the following statement is sometimes, always, or never true. justify your argument. 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 $x$.

Explanation:

Step1: Consider position of points

Let the coordinate of \(X\) be \(x\) and of \(Y\) be \(y\), and of \(W\) be \(w\). The distance from \(X\) to \(Y\) is \(|y - x|\). Given \(w=x+\frac{2}{5}|y - x|\).

Step2: Analyze cases

Case 1: If \(y>x\), then \(w=x+\frac{2}{5}(y - x)=\frac{3}{5}x+\frac{2}{5}y\), and \(w>x\).
Case 2: If \(y < x\), then \(w=x+\frac{2}{5}(x - y)=\frac{7}{5}x-\frac{2}{5}y\), and \(w>x\).

Answer:

Always true. Justification: Whether \(Y\) is to the right (\(y>x\)) or to the left (\(y < x\)) of \(X\) on the number - line, the coordinate of \(W\) which is \(\frac{2}{5}\) of the distance from \(X\) to \(Y\) away from \(X\) will be greater than the coordinate of \(X\).