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# Linear equations

### Solving inequations

• Simple versions of inequations can be solved or rearranged in ways similar to those used to solve equations.
• There are some differences:
• When you switch sides in an inequality you must reverse the sign. For example, $$2 < 3$$ must become $$3 > 2$$, otherwise it is not true.
• When you divide or multiply by a negative number you must reverse the inequality sign. For example, $$2 < 3$$, but when multiplied on both sides by $$-1$$ it must become $$-2 > -3$$, otherwise it is not true.
• For example, solve the following inequation for $$x$$:

$\newcommand{\eqncomment}{\scriptsize{\text{ #2}} } \newcommand{\ceqns}{ \begin{array}{rcll}} \newcommand{\ceqne}{\end{array}} \ceqns 1 - 2x &<& x + 2 \\ \\ 1 - 2x - x &<& x + 2 - x \ &\eqncomment{0.4}{subtract $$x$$ from both sides} \\ \\ 1 - 3x &<& 2 \\ \\ 1 - 3x - 1 &<& 2 - 1\ & \eqncomment{0.4}{subtract $$1$$ from both sides} \\ \\ -3x &<& 1 \\\\ \displaystyle \frac{-3x}{-3} & > & \displaystyle \frac{1}{-3} &\eqncomment{0.4}{divide both sides by $$-3$$} \\ \\ x &>& \displaystyle -\frac{1}{3} \ceqne$