91Ó°ÊÓ

Solving absolute value equations involves breaking down the problem into simpler parts. The absolute value of a number is its distance from zero on a number line, regardless of direction. So, the absolute value of both 4 and -4 is 4. To solve an absolute value equation like \(\bigg|\frac{2x-1}{3}\bigg| = 4\), we need to consider two separate cases. This is because the expression inside the absolute value can be either positive or negative.

Set notation is a way to specify a collection of objects or numbers. We use sets to list all possible solutions to an equation. For our exercise, once we have the separate solutions from each case, we combine them into a set. For example:
\[ x \in \textbackslash \{ \frac{13}{2}, \-\frac{11}{2} \textbackslash \}\]
Here, the symbol \( \in \) means 'belongs to', and the curly braces \( \textbackslash \{\} \) list the solutions. Each solution gets separated by a comma inside the braces. This notation clearly shows all possible values of \( x \) that satisfy the absolute value equation.

To solve \(\bigg|\frac{2x-1}{3}\bigg| = 4\), we take advantage of the definition of absolute value: an expression inside absolute value can be both positive and negative. Thus, we split it into two cases:
1. \(\frac{2x-1}{3} = 4\)
2. \(\frac{2x-1}{3} = -4\)

For the first case, solve the equation by removing the fraction, adding constants, and then isolating \( x \)
1. Multiply both sides by 3:
\[ 2x - 1 = 12 \]
2. Add 1 to both sides:
\[ 2x = 13 \]
3. Divide by 2:
\[ x = \frac{13}{2} \]

For the second case, follow similar steps:
1. Multiply both sides by 3:
\[ 2x - 1 = -12 \]
2. Add 1 to both sides:
\[ 2x = -11 \]
3. Divide by 2:
\[ x = -\frac{11}{2} \]

Combining these solutions gives our answer in set notation:
\[ x \in \textbackslash \{ \frac{13}{2}, -\frac{11}{2} \textbackslash \}\]