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### Section 2.14 : Absolute Value Equations

1. Solve the following equation.

$\left| {4p - 7} \right| = 3$

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Start Solution

There really isnâ€™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,

$4p - 7 = - 3\hspace{0.25in}{\mbox{or}}\hspace{0.25in}4p - 7 = 3$

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 3 is for the quantity inside to be either -3 or 3. 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*}4p & = 4 & \hspace{0.25in} & {\mbox{or}} & \hspace{0.25in}4p & = 10\\ p & = 1 & \hspace{0.25in} & {\mbox{or}} & \hspace{0.25in}p & = \frac{{10}}{4} = \frac{5}{2}\end{align*}

The two solutions are then : $$\require{bbox} \bbox[2pt,border:1px solid black]{{p = 1\,\,\,\,{\mbox{and }}p = \frac{5}{2}}}$$ .