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Lento vs. despacio

While both lento and despacio relate to the concept of slow, their usage is fundamentally different based on their grammatical function. The main distinction is that one is an adjective and the other is an adverb.

Lento

A1
Lento is an adjective. It is used to describe a noun (a person, thing, process, or concept) as being inherently slow. As an adjective, it must agree in gender and number with the noun it modifies (e.g., lento, lenta, lentos, lentas).
El servicio en ese restaurante es muy lento.
(The service at that restaurant is very slow.)
La tortuga es un animal lento.
(The turtle is a slow animal.)
La conexión a internet está muy lenta hoy.
(The internet connection is very slow today.)
Su recuperación de la gripe fue un proceso lento.
(His recovery from the flu was a slow process.)
Es una película con un ritmo lento pero profundo.
(It is a movie with a slow but profound pace.)
Las agujas del reloj parecían más lentas que nunca.
(The hands of the clock seemed slower than ever.)

Despacio

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Despacio is an adverb. It is used to describe a verb (an action), indicating that the action is performed in a slow manner or at a low speed. As an adverb, it is invariable, meaning it does not change its form.
Por favor, habla más despacio que no te entiendo.
(Please, speak more slowly as I don't understand you.)
El coche avanzaba despacio por el denso tráfico.
(The car moved slowly through the dense traffic.)
Camina despacio, el suelo está mojado.
(Walk slowly, the floor is wet.)
Mi abuela siempre come muy despacio.
(My grandmother always eats very slowly.)
Para aprender bien, tienes que leer el libro despacio.
(To learn well, you have to read the book slowly.)
El artista pintaba el cuadro despacio y con mucha precisión.
(The artist painted the picture slowly and with great precision.)

Summary

In essence, the difference is grammatical. Use lento (adjective) to describe *what* something is: a slow car (coche lento). Use despacio (adverb) to describe *how* an action is performed: to drive slowly (conducir despacio). Lento modifies nouns, while despacio modifies verbs.