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Inequalities

Inequalities on number line

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;

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)$

What inequality does this number line show?

Solution:

Remember,
A filled-in circle includes the number it is located on.
An open circle does not include the number it is located on.

We want to write an inequality that says x can be anything shown by the arrow and circle. The open circle located on 6 means that x cannot be equal to 6. The arrow pointing to the right means that x can be any number greater than 6.

Since x can be any number greater than 6, the inequality is x > 6 .

What inequality does this number line show?

Solution: $\fs4x$