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When faced with absolute value equations like \(|5b + 3| + 6 = 19\), the first step towards solving them is to isolate the absolute value on one side of the equation. This helps to simplify the problem.
In this exercise, begin by subtracting 6 from both sides: \(|5b + 3| = 19 - 6\). This leaves us with \(|5b + 3| = 13\). Understanding the properties of absolute value is crucial here. The absolute value function, represented by two vertical bars, measures the distance of a number from zero on the number line. It can never be negative, and it behaves in a piecewise manner, meaning that it splits the function into two separate cases based on whether the inside of the absolute value is positive or negative.
Thus, you will create two equations: one where the inside is positive \(5b + 3 = 13\) and another where it is negative \(5b + 3 = -13\). Each scenario is then solved separately for the variable \(b\).

Algebraic solutions involve finding the value of variables that satisfy the given equations. In our original exercise, solving both cases of the absolute value equation leads us to two different solutions.
In **Case 1**, where \(5b +3 = 13\), you subtract 3 from both sides to get \(5b = 10\). Then you divide both sides by 5 to isolate \(b\), giving \( b = 2 \). In **Case 2**, where \(5b + 3 = -13\), the process is similar. Subtracting 3 from both sides results in \(5b = -16\). After dividing both sides by 5, you reach \( b = \frac{-16}{5} \).
Both solutions are valid algebraic answers. This means that plugging these values back into the original equation should satisfy it, reinforcing the importance of checking your answers in algebra.

Getting to a solution using a step by step method ensures accurate and understandable results. Absolute value equations can be tricky because they involve two potential equations.
Initially, the absolute value needs to be isolated. Then, separate the equation into two logical cases, reflecting the dual nature of absolute values. Solve each resulting equation methodically.

• Subtract Constant: Start by simplifying both sides to disconnect constants from the absolute value.
• Create Separate Cases: Write two equations – one for the positive case and one for the negative case of the absolute value expression.
• Solve Each Equation: Use basic algebraic techniques to solve each equation and determine all possible solutions for the variable.

Following each step not only builds confidence but also assures a well-rounded understanding of the processes involved. This approach can be applied widely, ensuring all potential solutions are discovered and checked for correctness.