The triangle in the image is equilateral, which means all its sides are equal. The given side lengths are $4x - 5$ and $7$. To find $x$, we set the two expressions for the sides equal to each other:

$4x - 5 = 7$

Step-by-step Solution:

1. Add 5 to both sides: $4x = 12$

2. Divide both sides by 4: $x = 3$

Final Answer:

$x = 3$

Would you like me to explain further or clarify any steps?

Here are 5 related questions you could explore:

1. What are the properties of an equilateral triangle?
2. How do you find the perimeter of an equilateral triangle if all sides are expressed in terms of $x$?
3. How would the equation change if another side length was given instead of $7$?
4. How can we confirm that the triangle is indeed equilateral using angles?
5. How would the problem change if the triangle was isosceles instead of equilateral?

Tip: In equilateral triangles, all sides and angles are equal, so always use that to set equations when solving for variables.