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Inequalities

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;

To solve for a variable, use inverse operations to undo the operations in the inequality. Be sure to do the same operation to both sides 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.

Solve for y

 -7(y-10) + 2 > 9 -7(y-10) + 2 -2 > 9 - 2 Subtract 2 from both sides -7(y-10) > 7 Simplify

In the next step, you will divide both sides of the inequality by a negative number. Be sure to also reverse the direction of the inequality symbol.

 $\frac{-7(y-10)}{-7}=\frac{7}{-7}$ Divide both sides by - 7 and reverse the inequality symbol y – 10 < - 1 Simplify y – 10 + 10 < - 1 + 10 Add 10 to both sides y < 9 Simplify

Solve for w.

$\frac{w+10}{9}\;<\;-5$

Solution: w