Problem 41 - Algebra 2

Question

Solve each equation. Check your solutions. \(16 t=4|3 t+8|\)

Step-by-Step Solution

Answer

The solution to the equation is \(t = 8\).

Step 1: Understand the Equation

We are given the equation \(16t = 4|3t + 8|\). Our goal is to find the value(s) of \(t\) that satisfy this equation.

Step 2: Simplify the Equation

First, divide both sides by 4 to simplify the equation: \(\frac{16t}{4} = \frac{4|3t + 8|}{4}\). This simplifies to \(4t = |3t + 8|\).

Step 3: Analyze the Absolute Value

The equation \(4t = |3t + 8|\) can create two scenarios due to the absolute value.1. \(4t = 3t + 8\) (when \(3t + 8\) is positive or zero)2. \(4t = -(3t + 8)\) (when \(3t + 8\) is negative)

Step 4: Solve the First Case

For the first case, solve \(4t = 3t + 8\).Subtract \(3t\) from both sides: \[4t - 3t = 8\] This simplifies to \[t = 8\].

Step 5: Solve the Second Case

For the second case, solve \(4t = -(3t + 8)\).Distribute the negative sign:\[4t = -3t - 8\]. Add \(3t\) to both sides:\[4t + 3t = -8\] This simplifies to \[7t = -8\].Divide by 7:\[t = -\frac{8}{7}\].

Step 6: Check Solutions for Validity

For \(t = 8\):Plug back into the original equation:\[16(8) = 4|3(8) + 8|\] \[128 = 4|24 + 8|\] \[128 = 4(32)\] \[128 = 128\].For \(t = -\frac{8}{7}\):Plug back into the original equation:\[16\left(-\frac{8}{7}\right) = 4|3\left(-\frac{8}{7}\right) + 8|\] \[-\frac{128}{7} = 4|-\frac{24}{7} + \frac{56}{7}|\] \[-\frac{128}{7} = 4|\frac{32}{7}|\] \[-\frac{128}{7} = \frac{128}{7}\],which is not true.

Step 7: State the Valid Solution

Since only \(t = 8\) satisfies the original equation, \(t = -\frac{8}{7}\) is not valid. Hence, the solution is \(t = 8\).

Key Concepts

Understanding Absolute Value Equations

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.

Mastering Algebraic Manipulation

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.

The Importance of Checking Solutions

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.