Problem 31 - College Algebra with Corequisite Support

Question

For the following exercises, solve the equation involving absolute value. $$ |3 x-4|=8 $$

Step-by-Step Solution

Answer

The solutions are \(x = 4\) and \(x = -\frac{4}{3}\).

Step 1: Understand the Problem

The problem requires solving the equation \(|3x - 4| = 8\). This means finding values for \(x\) that make the absolute value expression true.

Step 2: Set Up Two Equations

The absolute value equation \(|3x - 4| = 8 \) can be split into two separate equations: 1. \(3x - 4 = 8\)2. \(3x - 4 = -8\)These equations represent the two scenarios where the absolute value yields 8 or -8.

Step 3: Solve the First Equation

Let's solve \(3x - 4 = 8\):Add 4 to both sides:\[ 3x = 12 \]Divide both sides by 3:\[ x = 4 \]

Step 4: Solve the Second Equation

Now solve \(3x - 4 = -8\):Add 4 to both sides:\[ 3x = -4 \]Divide both sides by 3:\[ x = -\frac{4}{3} \]

Step 5: Verify the Solutions

Verify both solutions by plugging them back into the original absolute value equation. For \(x = 4\):\(|3(4) - 4| = |12 - 4| = |8| = 8\)For \(x = -\frac{4}{3}\):\(3 \left(-\frac{4}{3}\right) - 4 = -4 - 4 = -8\) and \(|-8| = 8\)Both solutions satisfy the original equation.

Key Concepts

Equation Solving

Solving an equation is like finding the missing piece of a puzzle to make everything fit perfectly. In our case, we need to solve \(|3x - 4| = 8\). This means finding the values of \(x\) that make both sides equal when the absolute value is calculated. When dealing with absolute values, we understand that the expression inside can be either positive or negative and still result in the main value specified, which here is 8. This is why we create two separate equations: one where the inside of the absolute value is positive (\(3x - 4 = 8\)), and another where it is negative (\(3x - 4 = -8\)). By solving both equations, you can unlock the potential solutions for \(x\), allowing us to see under which conditions our absolute value equation holds true.

Algebraic Expressions

Algebraic expressions involve numbers, variables, and operations. They are like instructions telling us what to do with the values of the variables. For the equation \(|3x - 4| = 8\), the expression \(3x - 4\) is what sits inside the absolute value sign.When we manipulate algebraic expressions, we use operations like addition, subtraction, multiplication, and division to isolate the variable and solve for it. In this example, solving \(3x - 4 = 8\) involved:

• Adding 4 to get rid of the subtraction: \(3x = 12\)
• Dividing by 3 to isolate \(x\): \(x = 4\)

For \(3x - 4 = -8\), the process was similar:

• Add 4: \(3x = -4\)
• Divide by 3: \(x = -\frac{4}{3}\)

Handling these expressions requires a systematic approach to undo the operations and get the variable by itself.

Verification of Solutions

After solving the equations, verifying the solutions is the final but important step. We do this by substituting our solutions back into the original equation to check if they hold true.In this context, we received two values of \(x\): \(x = 4\) and \(x = -\frac{4}{3}\). Plugging \(x = 4\) back into the equation:\[ |3(4) - 4| = |12 - 4| = |8| = 8 \]The calculation confirms that \(x = 4\) is indeed a valid solution.Repeating this process for \(x = -\frac{4}{3}\):\[ 3\left(-\frac{4}{3}\right) - 4 = -4 - 4 = -8 \]and \[ |-8| = 8 \]The solutions satisfy the equation which confirms we solved it correctly.Verification is crucial as it assures us that both calculations and results accurately satisfy the conditions of the problem.