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$\overline{M L}$ is a median of $\triangle J K L .$ Which inequality best describes the range of values for $x ?$

(FIGURE CANNOT COPY)

(A) $x>2$

(B) $x>10$

(C) $3<x<4 \frac{2}{3}$

(D) $3<x<10$

$$D$$

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we need to find all possible values of X in the diagram. Three x minus nine is represents the angle measure of angle. K M L two x Plus one represents the angle measure of angle J M. L. And we are told that Segment ML is a median. Knowing that Segment ML is a median immediately applies that point. M is a midpoint of segment KJ, and if it is the midpoint of segment KJ, then that means segment K M and segment J M are congruent segments. If we see the media is dividing this triangle into two congruent or two triangles, then they both share segment ml, which means we've got two pairs of corresponding congrats int sides and the angle measures indicated are included angles. So the converse of the hinge there, um, says if the third side of one triangle is smaller than the third side of the other triangle than the included angle, opposite of the smaller side is gonna be a smaller angle. So this sets up an algebra statement of three X minus nine. Must be less than two X plus one and doing a couple of outburst steps. We're going to subtract two x that leaves X minus nine is less than one and adding nine to both sides. That gives us X is less than 10 but this is not the only situation we need to consider. Three X minus nine is the smaller side measurement, but it is the length of aside and it cannot equal zero. It cannot be less than zero. So we have a second condition that states three X minus nine must be greater than zero. So again, a couple algebra steps gives us three. X is greater than nine and dividing by three on both sides, we get X is greater than three. So the statement X is between the values of three and 10 is the same as saying three is less than X, which is also less than 10.

University of Oklahoma