Question

Solve: StartAbsoluteValue StartFraction x Over 6 EndFraction minus StartFraction 7 Over 12 EndFraction EndAbsoluteValue equals StartFraction 11 Over 6 EndFraction. =

Negative Direction:

– 3 lines of math labeled Positive Direction. The first line is: StartFraction x Over 6 EndFraction minus StartFraction 7 Over 12 EndFraction equals negative StartFraction 11 Over 6 EndFraction. The second line is: StartFraction x Over 6 EndFraction equals StartFraction 29 Over 12 EndFraction = –

= –2 lines of math labeled Negative Direction. The first line is: StartFraction x Over 6 EndFraction minus StartFraction 7 Over 12 EndFraction equals negative StartFraction 11 Over 6 EndFraction. The second line is: StartFraction x Over 6 EndFraction equals negative StartFraction 15 Over 12 EndFraction

Positive Direction:

– =

=



Determine the values for x by completing the steps shown.

x = x equals StartFraction 29 Over 2 EndFraction or x equals negative StartFraction 15 Over 2 EndFraction or x = –
x = –x equals negative StartFraction 29 Over 2 EndFraction or x equals StartFraction 15 Over 2 EndFraction or x =
x = –x equals negative StartFraction 29 Over 2 EndFraction or x equals negative StartFraction 15 Over 2 EndFraction or x = –
x = x equals StartFraction 29 Over 2 EndFraction or x equals StartFraction 15 Over 2 EndFraction or x =

Answer 1

To solve the equation

\[
\left| \frac{x}{6} - \frac{7}{12} \right| = \frac{11}{6},
\]

we break it down into two cases considering the definition of absolute value.

### Positive Direction

1. **Case 1**:

\[
\frac{x}{6} - \frac{7}{12} = \frac{11}{6}
\]

To solve for \( x \):

\[
\frac{x}{6} = \frac{11}{6} + \frac{7}{12}
\]

First, we need a common denominator to add the fractions. The least common multiple of 6 and 12 is 12.

Convert \(\frac{11}{6}\) to have a denominator of 12:

\[
\frac{11}{6} = \frac{22}{12}
\]

Now add:

\[
\frac{x}{6} = \frac{22}{12} + \frac{7}{12} = \frac{29}{12}
\]

Now, multiply both sides by 6:

\[
x = 6 \cdot \frac{29}{12} = \frac{174}{12} = \frac{29}{2}
\]

### Negative Direction

2. **Case 2**:

\[
\frac{x}{6} - \frac{7}{12} = -\frac{11}{6}
\]

To solve for \( x \):

\[
\frac{x}{6} = -\frac{11}{6} + \frac{7}{12}
\]

Again, convert \(-\frac{11}{6}\) to have a denominator of 12:

\[
-\frac{11}{6} = -\frac{22}{12}
\]

Now add:

\[
\frac{x}{6} = -\frac{22}{12} + \frac{7}{12} = -\frac{15}{12}
\]

Now, multiply both sides by 6:

\[
x = 6 \cdot -\frac{15}{12} = -\frac{90}{12} = -\frac{15}{2}
\]

### Solution:

Thus, the solutions for \( x \) are:

\[
x = \frac{29}{2} \quad \text{and} \quad x = -\frac{15}{2}
\]

Final answer presentation:

- \( x = \frac{29}{2} \)
- \( x = -\frac{15}{2} \)