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Inequalities
Multistep inequalities I

An inequality is an expression that involves an inequality sign (such as < , ≤ , > or ≥) instead of an equals sign.

Solving linear inequalities uses similar methods as solving multi-step equations (perform the same operation on both sides of the inequality), except that there are extra rules when using multiplication and division. An inequality remains unchanged if:

• the same number is added to both sides of the inequatily.
• the same number is subtracted from both sides of the inequality.
• both sides of the inequality are multiplied by a positive number.
• both sides of the inequality are divided by a negative number.

However, when you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality symbol.

Inequalities involving more than one operation can be solved by undoing the operations in reverse order in the same way you would solve an equation with more than one operation.

Solve for x:

$2(x+1)-7\;\le\;5(x-2)+8\;\Rightarrow\;2x+2-7\;\le\;5x-10+8\;\Rightarrow\;$

$2x-5x\;\le\;-10+8-2+7\;\Rightarrow\;-3x\;\le\;3\;\Rightarrow\;x\;\ge\;\frac{3}{-3}\;\Rightarrow\;x\;\ge\;-1$

$\text{Solution\;Set:}\;\{\;x\;\in\;\Re\;\text{,}\;x\;\ge\;-1\;\}\;\text{=}\;\[-1,\infty\)$

Solve for x:

$4(x+3)-x\;\ge\;-3(2-x)\;\Rightarrow\;4x+12-x\;\ge\;-6+3x\;\Rightarrow\;$

$3x+12\;\ge\;3x-6\;\Rightarrow\;12\;\ge\;-6$

$\text{Solution\;Set:}\;\Re\;$

 $6(-9x-7)-7\;\lt\;-99-8(9x-4)$ Solution: $\fs4x$ < ≤ > ≥