To solve for the length $x$ in the given right triangle, we can use trigonometric functions. The side opposite the given angle $68^\circ$ is 19, and the side adjacent to the angle $68^\circ$ is $x$.

We will use the tangent function, which is defined as:

$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$

For this problem:

$\tan(68^\circ) = \frac{19}{x}$

To solve for $x$, rearrange the equation:

$x = \frac{19}{\tan(68^\circ)}$

Now, let's calculate $x$.The length of $x$ is approximately $7.68$ units.

Would you like more details or have any questions about this solution?

Here are some related questions you might find interesting:

1. How would you calculate the hypotenuse of this triangle?
2. What is the other non-right angle in this triangle?
3. How does the cosine function relate to the sides of this triangle?
4. Can you solve for $x$ using the sine function instead?
5. What would be the effect on $x$ if the angle were changed to $45^\circ$?
6. How would the length of $x$ change if the opposite side were longer?
7. Can you find the area of this triangle?
8. What is the perimeter of this triangle?

Tip: Remember that in a right triangle, trigonometric functions like sine, cosine, and tangent are directly related to the angles and the sides of the triangle. This makes them very useful for solving for unknown sides or angles.