Algebra - Absolute Value Equations

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Section 2.14 : Absolute Value Equations

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2. Solve the following equation.

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

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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.

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