91Ó°ÊÓ

When tackling an equation that involves absolute values, the first step is crucial: isolating the absolute value expression. This equation currently reads as \(|3x - 5| + 4 = 5\). To isolate the absolute value, we need to move any terms not involving the absolute value away from it.
To achieve this, subtract 4 from both sides of the equation, simplifying it to \(|3x - 5| = 1\). Now, the absolute value term is on its own, which is exactly what we need to effectively deal with it.
Isolating the absolute value makes understanding the two potential scenarios clearer: one where the expression inside the absolute value equals the positive number and one where it equals the negative of that number.

Once the absolute value expression is isolated, the next step involves splitting the equation. This is because an absolute value equation of the form \(|A| = B\) means that the expression inside, \(A\), could either be \(B\) or \(-B\).
In our problem, \( |3x - 5| = 1 \) can be split into two separate equations:

• \(3x - 5 = 1\)
• \(3x - 5 = -1\)

This step is essential as it reflects the definition of absolute value, which signifies distance from zero, not direction. By splitting the equation, we are accounting for both possible values the expression can take inside the absolute value.

Now that we have two separate equations, we can solve each one as a linear equation.
The first equation from our split, \(3x - 5 = 1\):

• Add 5 to both sides to obtain \(3x = 6\).
• Then, divide both sides by 3, leading to \(x = 2\).

The second equation, \(3x - 5 = -1\):

• Add 5 to both sides to get \(3x = 4\).
• Divide each side by 3, resulting in \(x = \frac{4}{3}\).

Solving these linear equations provides the solutions to the original absolute value equation. With each step, we're applying simple algebra rules to unravel the possibilities embedded within the absolute value.