# Linear programming - Standard form with variable restricted from both sides

I have a pretty straightforward linear programming problem here:

$$maximize \hskip 5mm -x_1 + 2x_2 -3x_3$$

subject to

$$5x_1 - 6x_2 - 2x_3 \leq 2$$ $$5x_1 - 2x_3 = 6$$ $$x_1 - 3x_2 + 5x_3 \geq -3$$ $$1 \leq x_1 \leq 4$$ $$x_3 \leq 3$$

Convert to standard form.

what boggles me is how to substitute $$x_1$$ since it’s restricted from both sides and I can’t move forward in the problem until I figure it out...

I’m not asking for the whole standard form, just how to approach this one variable. :)

• It is like having two separate inequalities $x_1\geq 1$ and $x_1\leq 4$. Treat them as individual constraints and proceed with your method. – Michal Adamaszek Dec 2 '19 at 8:42

Answer to your problem: You can always split a two sided inequality to take two inequalities like how you take 2 equalities as 3 different ones.

Split $$a ≥ b ≥ c$$ into $$a ≥ c$$ , $$b ≥ c$$ and $$a ≥ b$$

Solution: $$x_1=\frac{6+2x_3}{5}$$ Plug in x everywhere

$$6−4x_2−2x_3≤2$$

$$27x_3−15x_2≥−21$$

$$x_3≤3$$

$$\frac{-1}{2}≤x_3$$

Take one on x and one on y.

To Max. $$\frac{-6+10x-17y}{5}$$, We see that x has to as high as possible and y as low as possible and priority goes to y as it contributes more to the value of the given expression.

So we take this vertex,

Which will be Intersection of $$27y-15x=-21$$ and $$6-4x-2y=2$$. I leave the rest to you.

The standard form should use the inequalities of the form $$h(\mathbf{x}) \le 0$$, where $$\mathbf{x} = (x_1, x_2, x_3)^\top$$. Hence, e.g. $$5x_1 - 2x_3 = 6$$ should be rewritten as follows: $$5x_1 - 2x_3 - 6 \le 0, \\ -5x_1 + 2x_3 + 6 \le 0.$$

Hope it is clear how to proceed with other constraints.