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### Section 2.14 : Absolute Value Equations

4. Solve the following equation.

$\left| {2x - 3} \right| = 4 - x$

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Hint : Just because the quantity outside of the absolute value bars in not a number does not mean this problem works any differently. Just remember to be careful with your answers!
Start Solution

Despite the fact that the quantity outside of the absolute value bars is not a positive number doesn’t mean that we can’t use the same process that we used in the first two problems.

Using the formula from the notes gives,

$2x - 3 = - \left( {4 - x} \right) = x - 4\hspace{0.25in}{\mbox{or}}\hspace{0.25in}2x - 3 = 4 - x$ Show Step 2

Now solving each these linear equations gives,

\begin{align*}2x - 3 & = x - 4 & \hspace{0.25in} & {\mbox{or}} & \hspace{0.25in}2x - 3 & = 4 - x\\ x & = - 1 & \hspace{0.25in} & {\mbox{or}} & \hspace{0.25in}3x & = 7\\ x & = - 1 & \hspace{0.25in} & {\mbox{or}} & \hspace{0.25in}x & = \frac{7}{3}\end{align*} Show Step 3

Now, because the quantity outside of the absolute value bars was not a positive constant we need to be careful with the answers we got in the previous step. It is possible that one or both are not in fact solutions to the original equation. So, we need to verify each of the possible solutions from the previous step by checking them in the original equation.

$x = - 1:\,\,\,\,\,\,\,\,\left| {2\left( { - 1} \right) - 3} \right|\mathop = \limits^? 4 - \left( { - 1} \right)\,\,\,\,\,\,\,\, \to \,\,\,\,\,\,\,\left| { - 5} \right|\mathop = \limits^? 5\,\,\,\,\,\,\, \to \,\,\,\,\,\,\,5 = 5\hspace{0.25in}{\mbox{OK}}$ $x = \frac{7}{3}:\,\,\,\,\,\,\,\,\left| {2\left( {\frac{7}{3}} \right) - 3} \right|\mathop = \limits^? 4 - \left( {\frac{7}{3}} \right)\,\,\,\,\,\,\,\, \to \,\,\,\,\,\,\,\left| {\frac{5}{3}} \right|\mathop = \limits^? \frac{5}{3}\,\,\,\,\,\,\, \to \,\,\,\,\,\,\,\frac{5}{3} = \frac{5}{3}\hspace{0.25in}{\mbox{OK}}$

Therefore, the two solutions to the original equation are then : $$\require{bbox} \bbox[2pt,border:1px solid black]{{x = - 1\,\,\,\,\,{\mbox{and}}\,\,\,\,\,x = \frac{7}{3}}}$$ .