I have been informed that on March 7th from 6:00am to 6:00pm Central Time Lamar University will be doing some maintenance to replace a faulty UPS component and to do this they will be completely powering down their data center.
Unfortunately, this means that the site will be down during this time. I apologize for any inconvenience this might cause.
Paul
February 18, 2026
Section 2.14 : Absolute Value Equations
5. Solve the following equation.
\[\left| {\frac{1}{2}z + 4} \right| = \left| {4z - 6} \right|\]Show All Steps Hide All Steps
This problem works identically to all the problems in this section. The only way the two absolute values can be equal is if the quantities inside them are the same value or the same value except for opposite signs. Doing this gives,
\[\frac{1}{2}z + 4 = - \left( {4z - 6} \right) = 6 - 4z\hspace{0.25in}{\mbox{or}}\hspace{0.25in}\frac{1}{2}z + 4 = 4z - 6\]In other words, we can use the formula discussed in this section to do this problem!
Show Step 2Now solving each these linear equations gives,
\[\begin{align*}\frac{1}{2}z + 4 & = 6 - 4z & \hspace{0.25in} & {\mbox{or}} & \hspace{0.25in}\frac{1}{2}z + 4 & = 4z - 6\\ \frac{9}{2}z & = 2 & \hspace{0.25in} & {\mbox{or}} & \hspace{0.25in} - \frac{7}{2}z & = - 10\\ z & = \frac{4}{9} & \hspace{0.25in} & {\mbox{or}} & \hspace{0.25in}z & = \frac{{20}}{7}\end{align*}\]Now, because both sides of the equation have absolute values, we know that regardless of the value of \(x\) we plug into the original equation the absolute value will guarantee that the result will be positive and so we don’t need to verify either of these solutions.
Therefore, the two solutions to the original equation are then : \(\require{bbox} \bbox[2pt,border:1px solid black]{{z = \frac{4}{9}\,\,\,\,\,{\mbox{and}}\,\,\,\,\,z = \frac{{20}}{7}}}\) .