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How to show that the following two optimization problems are equivalent?

The first problem is $$\max. ~~g(\lambda)-x$$ $$s.t. ~~ x\geq 0, \lambda \geq 0, a-\lambda b +x \geq 0$$ where $g(\lambda)$ is an increasing function of $\lambda$ and $a,b$ are some positive constants and the optimization variables are $x,\lambda$.

I want to know how the above problem is equivalent to the following problem $$\min. ~~g(\lambda)+x$$ $$s.t. ~~ x\geq 0, \lambda \geq 0, x \geq a-\lambda b.$$ At this moment, I only understand the change in sign before $x$ in the objective function of second optimization problem. But I do not understand the rest of the changes. Any help in this regard will be much appreciated.

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