Question

analyze determine whether the following statement is sometimes, always, or never true. justify your argument. if (xy) is on a number - line and point (w) is (\frac{1}{3}) of the distance from (x) to (y), then the coordinate of point (w) is greater than the coordinate of point (x). select choice ; if the coordinate of (x) is 0 and the coordinate of (y) is negative, then the coordinate of (w) will be select choice and select choice than select choice. 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 than of (x). sometimes always never

Explanation:

Step1: Consider case 1

Let \(x = 0\) and \(y<0\). If \(W\) is \(\frac{2}{3}\) of the distance from \(X\) to \(Y\), the coordinate of \(W=x+\frac{2}{3}(y - x)=\frac{2y + x}{3}\). Substituting \(x = 0\), we get \(W=\frac{2y}{3}<0=x\).

Step2: Consider case 2

Let \(x>0\) and \(y>x\). The coordinate of \(W=x+\frac{2}{3}(y - x)=\frac{2y + x}{3}\). Since \(y>x\), \(\frac{2y + x}{3}>x\) because \(\frac{2y + x}{3}-x=\frac{2y + x - 3x}{3}=\frac{2(y - x)}{3}>0\).

Answer:

Sometimes