Question

How to solve a system of inequalities without graphing $?$

Answer 1

Solve a system of inequalities without graphing:

Elimination or Substitution of variables is not possible to solve the inequalities.

So, form a range for dependent variable using the independent variable.

Lets consider a following system of inequalities:

$$\begin{gathered} 3\mathrm{x}+2\mathrm{y}\le 5 \\ \mathrm{y}+\mathrm{x}\ge 1 \end{gathered}$$

[Make $\mathrm{y}$ as dependent variable to make its range with the equation of $\mathrm{x}$]

Operate on the first equation:

$$\begin{gathered} 3x+2y\le 5 \\ 2y\le 5-3x \end{gathered}$$ [Subtract $3\mathrm{x}$ from both side of the equation]

$\mathbf{y}\mathbf{\le }\frac{\mathbf{5}\mathbf{-}\mathbf{3}\mathbf{x}}{\mathbf{2}}$ [Divide both side of the equation by $2$]

Operate on the second equation:

$\mathrm{y}+\mathrm{x}\ge 1$

$\mathbf{y}\mathbf{\ge }\mathbf{1}\mathbf{-}\mathbf{x}$ [Subtract $\mathrm{x}$ from both side of the equation]

Therefore, the range of $\mathrm{y}$ will be as follows:

$\mathit{y}\mathbf{\in }\mathbf{[}\mathbf{1}\mathbf{-}\mathbf{x}\mathbf{,}\frac{\mathbf{5}\mathbf{-}\mathbf{3}\mathbf{x}}{\mathbf{2}}\mathbf{]}$

Hence, the range can be formed for the dependent variable using the independent variable.