Technology can also be used to solve absolute value equations graphically. For more specific instructions, consult your calculator’s manual or enter “solving absolute value equations using [enter the name of your graphing calculator or program]” into a search engine. More specific instructions for TI-83™ and TI-84™ calculators are provided in the Calculator Guide. If you have difficulty solving absolute value equations using technology, contact your teacher.

 Example  2
Solve \(\left|3x^2 - 2x -1 \right| = 2x + 1\) graphically, using technology.

Graph the functions \(f(x) = \left|3x^2 - 2x - 1 \right| \) and \(g(x) = 2x + 1\).



The points of intersection are \((0, 1)\), and approximately \((–0.39, 0.23)\) and \((1.72, 4.44)\). The solutions to this equation are \(0\), and approximately \(–0.39\) and \(1.72\).

Verify for \(x = 0\).

Left Side Right Side
\(\begin{array}{r}
 \left| {3x^2 - 2x - 1} \right| \\
 \left| {3\left( 0 \right) - 2\left( 0 \right) - 1} \right| \\
 \left| { - 1} \right| \\
 1  \end{array}\)
 \(\begin{array}{l}
 2x + 1 \\
 2\left( 0 \right) + 1  \\
1 \end{array}\)
\(\hspace{28pt}\)LS = RS

Verify for \(x = -0.39\).

Left Side Right Side
\(\begin{array}{r}
 \left| {3x^2 - 2x - 1} \right| \\
 \left| {3\left( { - 0.39} \right)^2 - 2\left( { - 0.39} \right) - 1} \right| \\
 \left| {0.2363} \right| \\
 0.2363 \end{array}\)
 \(\begin{array}{l}
 2x + 1 \\
 2\left( { - 0.39} \right) + 1 \\
 0.22 \end{array}\)
\(\hspace{47pt}\)LS \(\doteq\) RS

Verify for \(x = 1.72\).

Left Side Right Side
\(\begin{array}{r}
 \left| {3x^2 - 2x - 1} \right| \\
 \left| {3\left( {1.72} \right)^2 - 2\left( {1.72} \right) - 1} \right| \\
 \left| {4.4352} \right| \\
 4.4352  \end{array}\)
 \(\begin{array}{l}
 2x + 1 \\
 2\left( {1.72} \right) + 1 \\
 4.44  \end{array}\)
\(\hspace{45pt}\)LS \(\doteq\) RS