Question

How to solve a system of inequalities without graphing?

Answer 1

Hint: Inequalities do not allow for methods like the elimination of variables and also, they don’t have a single result but have a range and as such, take a variable which is the dependent variable and form its range by using the independent variable.

Complete step by step solution:
When we work with inequalities, doing a simple elimination of variables or substitution is not possible and as such, we need to form a range for the dependent variable using the independent variable.
Let us consider a system of inequalities.
$\
  3x + 5y \leqslant 7...............\left( i \right) \\
  y + 2x \geqslant 2.................\left( {ii} \right) \\
\ $
Now what we have to do here is take a dependent variable and make the other variable as its range. By convention, we take ‘y’ as the dependent variable. So now we will work with the first equation to make one side devoid of anything but ‘y’.
$\
  3x + 5y \leqslant 7 \\
  5y \leqslant 7 - 3x \\
  y \leqslant \dfrac{{7 - 3x}}{5} \\
\ $
Here we add 3x to both sides of the equation and then divide both sides by 5. This is done to make one side only have ‘y’.
We operate similarly on the second equation.
$\
  y + 2x \geqslant 2 \\
  y \geqslant 2 - 2x \\
\ $
Now that we have a value of ‘y’ which is dependent on ‘x’, we make its range with the equations of ‘x’.
The range will be $y \in \left[ {\dfrac{{7 - 3x}}{5},2 - 2x} \right]$. Now all we need to do is to place the values of x and y will be found out.

Note:
We need to have one greater than $\left( \geqslant \right)$ and one lesser than $\left( \leqslant \right)$ relation between y and x so that we can form the range. If it is not there, try to make it. If it is impossible to make then, think of which equation will give the lower value if both are in greater than form, and then the range will be that function to infinity. Similarly, if both equations are in less than form, choose one with the greater result and the range will be negative infinity to that function.