Example 2
Completion requirements
| Example 2 |
Solve the inequality \(x^2 -6x + 5 \gt 0\). Represent the solution set symbolically and on a number line.
Although grid paper or a graphing calculator can be used to graph the function in detail, the zeros and the basic shape of the graph of the function are all that is needed to solve the inequality.
Next, use the graph to determine the \(x\)-values for which \(x^2 - 6x + 5 \gt 0\).
Sometimes students will try to represent a two interval solution region, such as \(x \lt 1\) or \(x \gt 5\), using the notation \(1 \gt x \gt 5\), but this is incorrect. The notation \(1 \gt x \gt 5\) represents all \(x\)-values that are less than \(1\) and greater than \(5\), but no number will fit this criteria.
The corresponding function \(f(x) = x^2 - 6x + 5\) can be used to graphically solve the inequality. The solution regions are separated by the zeros of the function.
\(\begin{align}
x^2 - 6x + 5 &= 0 \\
\left( {x - 1} \right)\left( {x - 5} \right) &= 0
\end{align}\)
The coefficient on \(x^2\) is positive, so the parabola will open up. Use the direction of opening and the zeros to sketch the graph.
\(\begin{align}
x^2 - 6x + 5 &= 0 \\
\left( {x - 1} \right)\left( {x - 5} \right) &= 0
\end{align}\)
\(\begin{align}
x - 1 &= 0 \\
x &= 1
\end{align}\)
x - 1 &= 0 \\
x &= 1
\end{align}\)
\(\begin{align}
x - 5 &= 0 \\
x &= 5
\end{align}\)
x - 5 &= 0 \\
x &= 5
\end{align}\)
The coefficient on \(x^2\) is positive, so the parabola will open up. Use the direction of opening and the zeros to sketch the graph.
Although grid paper or a graphing calculator can be used to graph the function in detail, the zeros and the basic shape of the graph of the function are all that is needed to solve the inequality.
Next, use the graph to determine the \(x\)-values for which \(x^2 - 6x + 5 \gt 0\).
If \(x \lt 1\) or \(x \gt 5\), \(x^2 - 6x + 5 \gt 0\), so the solution set is {\(x | x \lt 1 \thinspace {\rm{or}} \thinspace x \gt 5, \thinspace x \in \rm{R}\)}.
Plot the solution set on a number line.
Plot the solution set on a number line.
Sometimes students will try to represent a two interval solution region, such as \(x \lt 1\) or \(x \gt 5\), using the notation \(1 \gt x \gt 5\), but this is incorrect. The notation \(1 \gt x \gt 5\) represents all \(x\)-values that are less than \(1\) and greater than \(5\), but no number will fit this criteria.