Algebra - Absolute Value Equations

Show Mobile Notice Show All Notes Hide All Notes

Mobile Notice

You appear to be on a device with a "narrow" screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best viewed in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (you should be able to scroll/swipe to see them) and some of the menu items will be cut off due to the narrow screen width.

Section 2.14 : Absolute Value Equations

Back to Problem List

2. Solve the following equation.

\[\left| {2 - 4x} \right| = 1\]

Show All Steps Hide All Steps

Start Solution

There really isn’t all that much to this problem. All we need to do is use the formula we discussed in the notes for this section. Doing that gives,

\[2 - 4x = - 1\hspace{0.25in}{\mbox{or}}\hspace{0.25in}2 - 4x = 1\]

Do not make the common mistake of just turning every minus sign inside the absolute value bars into a plus sign. That is just not how these work. The only way for the value of the absolute value to be 1 is for the quantity inside to be either -1 or 1. In other words, we get rid of the absolute value bars by using the formula from the notes.

Show Step 2

At this point all we need to do is solve each of the linear equations we got in the previous step. Doing that gives,

\[\begin{align*} - 4x & = - 3 & \hspace{0.25in} & {\mbox{or}} & \hspace{0.25in} - 4x & = - 1\\ x & = \frac{3}{4} & \hspace{0.25in} & {\mbox{or}}& \hspace{0.25in}x & = \frac{1}{4}\end{align*}\]

The two solutions are then : \(\require{bbox} \bbox[2pt,border:1px solid black]{{x = \frac{1}{4}\,\,\,\,{\mbox{and }}x = \frac{3}{4}}}\) .