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Absolute value represents the non-negative distance of a number from zero on a number line. This concept is crucial to understand in order to solve absolute value equations effectively. For instance, the absolute value of both 3 and -3 is 3, because each is three units away from zero.
When dealing with an equation like \( |2x - 3| = 8 \), the expression inside the absolute value brackets can be either positive or negative. This means we need to consider both situations to solve the equation thoroughly.

To solve an absolute value equation, you must split the original equation into two separate linear equations. This is because the absolute value equation \( |2x - 3| = 8 \) translates to two scenarios:

• The expression inside the absolute value is equal to 8 (\( 2x - 3 = 8 \)).
• The expression inside the absolute value is equal to -8 (\( 2x - 3 = -8 \)).

By setting up these two separate linear equations, you eliminate the absolute value and are able to solve for the variable efficiently.

When you have split the equations, you need to solve each linear equation individually. Let's start with the first one:
1. \(2x - 3 = 8 \)
To solve this, add 3 to both sides:
\(2x = 11 \).
Finally, divide by 2:
\(x = \frac{11}{2} \).

Next, solve the second linear equation:
2. \(2x - 3 = -8 \)
Add 3 to both sides:
\(2x = -5 \).
Then, divide by 2:
\(x = -\frac{5}{2} \).
These steps help you isolate the variable in each equation, providing the solutions for the original absolute value equation.

The concept of absolute value is directly linked to the idea of distance on a number line. Consider the original equation \( |2x - 3| = 8 \). In this context, the absolute value represents the distance of the expression \( 2x - 3 \) from zero.
By solving the equation, we find the points on the number line where the distance from zero equals 8. Thus, our solutions, \( x = \frac{11}{2} \) and \( x = -\frac{5}{2} \), give us the specific values at which the expression inside the absolute value is precisely 8 units away from zero.
Understanding this helps to visualize why we need to split the original equation into two and provides a deeper comprehension of the underlying mathematical principles.