The problem's introduction's description of the wave, which is going in the +x direction, is accurate.
Due to the particles just on left side of the wave going upward and approaching their peak, the wave is traveling to the left. Alternatively, if the particles just on right are traveling up and the particles just on left are moving down, the wave is moving to the right.
When a wave's disturbance travels perpendicular to the wave's direction, the wave is said to be transverse. While the disturbance swings up and down, you might imagine the wave going from left to right. A water wave with up-and-down water movement as it travels across the ocean is an illustration of a transverse wave.
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