Linear Inequalities: Linear Inequalities
Multiple linear inequalities with one unknown
In a system of inequalities with unknown #x# a solution is formulated by giving a description of all values of #x# for which all inequalities in the system are satisfied.
The system \[\lineqs{7x-3 &\ge& 0\cr 6x+5&\le&0\cr}\] is therefore not different from
\[\left(7x-3 \ge 0 \right) \land\left(6x+5\le0\right)\tiny.\]
How can you solve a system of linear inequalities with one unknown?
Solution by reduction
- First solve all linear inequalities individually.
- Than compose the results using the operator #\land# and simplify if possible.
Solution through equations
- First solve the equation at every inequality individually.
- Than see for each solution, and each of the line segments on the number line between two individual solutions of the equations, if the inequalities are satisfied.
#none#
After all, the solution of two individual inequality gives \[\lineqs{x &\ge& {{3}\over{4}} \cr x&\le&-{{3}\over{7}}\cr}\]
Or: #x \ge {{3}\over{4}} \land x\le -{{3}\over{7}}#. Because #{{3}\over{4}} \gt -{{3}\over{7}}#, there are no solutions. The answer is #none# .
After all, the solution of two individual inequality gives \[\lineqs{x &\ge& {{3}\over{4}} \cr x&\le&-{{3}\over{7}}\cr}\]
Or: #x \ge {{3}\over{4}} \land x\le -{{3}\over{7}}#. Because #{{3}\over{4}} \gt -{{3}\over{7}}#, there are no solutions. The answer is #none# .
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