# 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}[2]{\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

\]