Diophantine inequalities with mixed powers

It is shown that if λ₁,…, λ₆ are nonzero real numbers, not all of the same sign, such that $\text{λ}{}_1\text{λ}{}_2$ is irrational, then the values taken by λ₁x₁² + λ₂x₂² + λ₃x₃³ + λ₄x₄³ + λ₅x₅⁵ + λ₆x₆⁵ for integral x₁,…, x₆ are everywhere dense on the real line. Similar results are proved with other combinations in place of the two fifth powers.