<s>[INST] Solve $\sqrt[3]{7x+19}+\sqrt[3]{7x-19}=\sqrt[3]{2}$ by algebraic methods I was trying to solve this equation without using calculus.
Is it possible to be solved by elementary algebraic methods?
$$\sqrt[3]{7x+19}+\sqrt[3]{7x-19}=\sqrt[3]{2}$$
 [/INST] powering by $3$ we get 
$$7x+19+3\sqrt[3]{7x+19}^2\sqrt[3]{7x-19}+3\sqrt{7x+19}(\sqrt[3]{7x-19})^2+7x-19=2$$
and we get
$$14x+3\sqrt[3]{7x+19}\sqrt[3]{7x-19}(\sqrt[3]{7x+19}+\sqrt[3]{7x-19})=2$$
$$14x+3\sqrt[3]{7x+19}\sqrt[3]{7x-19}\sqrt[3]{2}=2$$
can you proceed?
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