Questions: Solve. 3/4 + 2x = 1/6

Solve. 3/4 + 2x = 1/6
Transcript text: Solve. \[ \left|\frac{3}{4}+2 x\right|=\frac{1}{6} \]
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Solution

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Solution Steps

To solve the equation \(\left|\frac{3}{4}+2x\right|=\frac{1}{6}\), we need to consider the two cases for the absolute value. The expression inside the absolute value can be either positive or negative. Therefore, we set up two separate equations: \(\frac{3}{4} + 2x = \frac{1}{6}\) and \(\frac{3}{4} + 2x = -\frac{1}{6}\). We then solve each equation for \(x\).

Step 1: Set Up the Equations

To solve the equation \(\left|\frac{3}{4} + 2x\right| = \frac{1}{6}\), we consider the two cases for the absolute value:

  1. \(\frac{3}{4} + 2x = \frac{1}{6}\)
  2. \(\frac{3}{4} + 2x = -\frac{1}{6}\)
Step 2: Solve the First Equation

For the first equation \(\frac{3}{4} + 2x = \frac{1}{6}\), we solve for \(x\):

\[ 2x = \frac{1}{6} - \frac{3}{4} \]

\[ 2x = \frac{1}{6} - \frac{9}{12} \]

\[ 2x = \frac{2 - 9}{12} = -\frac{7}{12} \]

\[ x = -\frac{7}{24} \approx -0.2917 \]

Step 3: Solve the Second Equation

For the second equation \(\frac{3}{4} + 2x = -\frac{1}{6}\), we solve for \(x\):

\[ 2x = -\frac{1}{6} - \frac{3}{4} \]

\[ 2x = -\frac{1}{6} - \frac{9}{12} \]

\[ 2x = -\frac{2 + 9}{12} = -\frac{11}{12} \]

\[ x = -\frac{11}{24} \approx -0.4583 \]

Final Answer

\(\boxed{x = -\frac{7}{24}, -\frac{11}{24}}\)

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