I need to make the output of a variable positive, for both negative and positive inputs

Question

I need to make the output of a variable positive, for both negative and positive inputs.

Since I am using it in the linear optimization problem, no squares, self-division or self-multiplication is allowed.

Need a mathematical solution, No prebuilt functions of programming languages are allowed

Answer 1

$$|5 x_1-7x_2| \leq 10$$ is equivalent to the two linear constraints $$5 x_1-7x_2 \leq 10 \text{ and } -(5 x_1-7x_2) \leq 10$$

Answer 2

Let the input value be $x$. Then compare $x$ with $0$, if it is greater than $0$ we are done. If it is less than $0$, return $-x$.

Answer 3

Define a function $$f(x)=\mid x\mid=\left\{\begin{array}{ll}x&\text{if }x>0\\-x&\text{otherwise}\end{array}\right..$$