What do you call an equation that is impossible to algebraically solve?

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For example...

$$3.5=\sin{(x)}+\frac{\cos{(x)}}{\ln{(x)}+\sin{(x)}}$$

So your options to solving such a thing are iteration and other numerical methods. Does an equation that's not solvable algebraically have a specific name?

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These are called transcendental equations. When you have a polynomial equation that you cannot solve, then you say that the equation "is not solvable by radicals."