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When you encounter an absolute value equation like \(16t = 4|3t + 8|\), it is crucial to understand that the absolute value affects how you approach solving the problem. The absolute value \(|x|\) signifies the 'distance' of \(x\) from zero on the number line, meaning it always results in a non-negative value. This introduces two different scenarios into your equation:

• The expression inside the absolute value is non-negative, i.e., \(3t + 8 \geq 0\)
• The expression inside the absolute value is negative, i.e., \(3t + 8 < 0\)

Taking these two scenarios helps us create two linear equations to solve separately. The absolute value property allows transformation into these meaningful statements to see how \(t\) behaves. This method of handling absolute value equations lets you explore all potential solutions in detail.

Algebraic manipulation is the toolset you'll need to solve the linear equations derived from our absolute value problem. For instance, after simplifying the original equation to \(4t = |3t + 8|\), you break it down into two parts:

• \(4t = 3t + 8\), which when rearranged becomes \(t = 8\)
• \(4t = -(3t + 8)\), which when simplified gives \(7t = -8\) and further processing results in \(t = -\frac{8}{7}\)

By isolating the variable \(t\), you move through arithmetic operations like addition, subtraction, multiplication, and division to rearrange and simplify. Proper use of these operations helps isolate the variables and reach potential solutions quickly and accurately.

After solving the absolute value equation, checking your solutions is a crucial step. This ensures that every proposed solution is valid for the original equation. For our solution \(t = 8\), substituting back into the initial equation \(16t = 4|3t + 8|\) shows:\[16(8) = 4|3(8) + 8|\] translates to \[128 = 128\], confirming that \(t = 8\) works.For \(t = -\frac{8}{7}\), the substitution leads to an inconsistency:\[16\left(-\frac{8}{7}\right) = 4|-\frac{24}{7} + \frac{56}{7}|\] becomes untrue as \[-\frac{128}{7} eq \frac{128}{7}\].Thus, validation through substitution differentiates between feasible and infeasible solutions. Only \(t = 8\) is valid, underscoring this critical checking step. Verification prevents errors and assures accurate completion of equation-solving tasks.