Inequalities

Tags

math

An inequality is an expression that expresses a relation between two numbers. For example \( 5 < 10 \) is an inequality.

Warn: When multiplying an inequality by a negative sign, you must reverse the direction of the inequality. For example \( 5 < 10 \rightarrow -5 > -10 \).

Solving Quadratic Inequalities

You may sometimes be asked to find the range of values for which a formula, such as a quadratic equation, is within some range. For example find the values of \( x \) for which the following is true

\begin{align} \label{eq:quadratic-inequality-eg} x^2 - 4x - 5 < 0 \end{align}

To do so you must first solve the quadratic and find its real valued roots, in this case \( (x+1)(x-5) = 0 \). The roots here shows that the quadratic equation meets the x-axis at \( x = -1 \) and \( x = -5 \). The only way for the curve to become negative or positive is if it meets the curve at some point, therefore we can simply check the position of the curve between its intersection with the $x$-axis to see whether its negative or not. We don't know which direction the curve is travelling when it meets this axis so we have to either plot or substitute values in the range to determine the trajectory of the curve and then assert whether its within or outside the inequality we want.

Using a Plot

\begin{figure}
  \centering
  \begin{tikzpicture}
    \begin{axis}[axis lines=center, domain=-10:10]
      \addplot[color=red]{x^2 - 4*x - 5};,
      \addplot[only marks] coordinates {(-1, 0) (5, 0)};,
    \end{axis}
  \end{tikzpicture}
  \caption{Plot of equation \autoref{eq:quadratic-inequality-eg}}
  \label{fig:quadratic-inequality-eg}
\end{figure}

In fig:quadratic-inequality-eg we see that the curve dips below \(0\) when entering the root \( x = -1 \) and rises back to above \(0\) when entering meeting the root \( x = 5 \). Therefore the solution to eq:quadratic-inequality-eg is \( -1 < x < 5 \).

Using a Number Line

Alternatively we could've created a number-line and substituted in values for \(x\) before, within and after roots to see how the curve shifts.

\begin{table}[] \centering \begin{tabular}{c|ll|lllll|ll}

Sign & \multicolumn{2}{c|}{+ve} & \multicolumn{5}{c|}{-ve} & \multicolumn{2}{c}{+ve} \\\\
\hline
x    & -2         & -1          & 0   & 1  & 2  & 3  & 4   & 5          & 6          \\\\
y    & 7          &             & -5  &    &    &    &     &            & 7

\end{tabular} \label{tbl:quadratic-inequality-eg} \caption{Number line showing how the value of $y$ changes across the x-axis intersections.} \end{table}

In tbl:quadratic-inequality-eg we see that at \( x = -2 \) the curve is positive, at \( x = 0 \) it's negative and at \( x = 6 \) it's again positive. This leads to the same conclusion that \( -1 < x < 5 \).