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Solo vs. soltero

While both solo and soltero can relate to the idea of being single in English, they are not interchangeable in Spanish. Solo describes the temporary state of being physically alone or can mean only, whereas soltero specifically refers to the official marital status of being unmarried.

Solo

A1
The word solo is primarily used in two ways. As an adjective, it means alone, by oneself, or lonely, describing a temporary situation of not being with other people. It must agree in gender and number with the noun it describes (sola, solos, solas). As an adverb, it means only or just.
Anoche me quedé solo en casa viendo una película.
(Last night I stayed home alone watching a movie.)
Ella prefirió viajar sola para tener más libertad.
(She preferred to travel alone to have more freedom.)
A veces me siento un poco solo los domingos por la tarde.
(Sometimes I feel a little lonely on Sunday afternoons.)
No me gusta el café con leche, lo tomo solo.
(I don't like coffee with milk, I drink it black (alone).)
Solo quiero un minuto de tu tiempo.
(I only want a minute of your time.)
Esto es solo el principio de nuestra aventura.
(This is just the beginning of our adventure.)
Los niños no pueden ir solos al parque.
(The children cannot go to the park by themselves.)

Soltero

A2
The word soltero is an adjective or noun that describes a person's marital status. It means single in the sense of being unmarried. It is a more formal, long-term, or legal state than solo. It must also agree in gender and number (soltera, solteros, solteras).
Mi hermano tiene 35 años y sigue soltero.
(My brother is 35 years old and is still single.)
En el formulario debes indicar si eres casado o soltero.
(On the form, you must indicate if you are married or single.)
Ella es una mujer fuerte, independiente y soltera.
(She is a strong, independent, and single woman.)
La fiesta estaba llena de jóvenes solteros y solteras.
(The party was full of young single men and women.)
Él disfruta mucho su vida de soltero.
(He really enjoys his life as a single man.)
Mi amiga es madre soltera de dos niños.
(My friend is a single mother of two children.)
Compró un apartamento de soltero en el centro de la ciudad.
(He bought a bachelor pad in the city center.)

Summary

In essence, use solo (or sola, solos, solas) to describe the state of being physically alone or lonely, or as an adverb for only. Use soltero (or soltera, solteros, solteras) to describe the marital status of being unmarried. A person can be soltero (unmarried) but at a party with friends, so they are not solo (alone).