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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}[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
\]