Multiple linear inequalities with one unknown

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.

Solve #x# in #\lineqs{{{8\cdot x}\over{3}}-2 &\ge& 0\cr {{15\cdot x}\over{2}}+{{21}\over{2}}&\le&0\cr}#

Give your answer in the form of inequalities and enter #none# if there is no solution.

#none#

After all, the solution of two individual inequality gives \[\lineqs{x &\ge& {{3}\over{4}} \cr x&\le&-{{7}\over{5}}\cr}\]
Or: #x \ge {{3}\over{4}} \land x\le -{{7}\over{5}}#. Because #{{3}\over{4}} \gt -{{7}\over{5}}#, there are no solutions. The answer is #none# .

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