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Inequalities

# Linear inequalities in one variable

A statement involving a variable and a sign of inequality (viz. < , ≤ , > or ≥) is called an inequality. A statement of inequality between two expressions consisting of a single variable, say x, of highest power 1, is called a linear inequality in one variable. It is ussually written in any of the following forms:

ax+b<0
ax+b>0
ax+b≥0
ax+b≤0

where a ≠ 0;

You can solve an inequality in the variable x by finding all values of x for which the inequality is true. Such values are solutions and are said to satisfy the inequality. The solution set of an inequal¡ty is the set of all real numbers that are solutions of the inequality.

Linear inequalities are solved much the same way as linear equations are solved, with one important exception: when multiplying or dividing both sides of an inequality by a negative number, the inequality sign must be reversed.

$2x-5\;<\;3\;\Rightarrow\;2x\;<\;3+5\;\Rightarrow\;2x\;<\;8\;\Rightarrow\;x\;<\;4$

$\text{Solution\;Set:}\;\{\;x\;\in\;\Re\;\text{,}\;x\;<\;4\;\}\;\text{=}\;$$-\infty,4)$ $2x+1\;\le\;5x-8\;\Rightarrow\;2x-5x\;\le\;-8-1\;\Rightarrow\;-3x\;\le\;-9\;\Rightarrow\;x\;\ge\;3$ $\text{Solution\;Set:}\;\{\;x\;\in\;\Re\;\text{,}\;x\;\ge\;3\;\}\;\text{=}\;\[3,\infty$$$

 $7x-5\;\ge\;4x+19$ Solution: $\fs4x$ < ≤ > ≥