Problem 35 - Intermediate Algebra

Question

Solve each equation. See Example 2. $$ |3-4 x|+1=6 $$

Step-by-Step Solution

Answer

The solutions are \( x = -\frac{1}{2} \) and \( x = 2 \).

Step 1: Isolate the Absolute Value

First, subtract 1 from both sides of the equation to isolate the absolute value expression. \[ |3 - 4x| + 1 = 6 \] Subtracting 1 gives: \[ |3 - 4x| = 5 \]

Step 2: Set Up Two Separate Equations

Since the absolute value of a number can be equal to a positive or negative value, set up two separate equations to address both possibilities:1. \( 3 - 4x = 5 \)2. \( 3 - 4x = -5 \)

Step 3: Solve the First Equation

Solve \( 3 - 4x = 5 \):1. Subtract 3 from both sides: \[ -4x = 2 \]2. Divide by -4: \[ x = -\frac{1}{2} \]

Step 4: Solve the Second Equation

Solve \( 3 - 4x = -5 \):1. Subtract 3 from both sides: \[ -4x = -8 \]2. Divide by -4: \[ x = 2 \]

Step 5: Verify the Solutions

Substitute \( x = -\frac{1}{2} \) and \( x = 2 \) back into the original equation to ensure they satisfy it.1. For \( x = -\frac{1}{2} \): \[ |3 - 4(-\frac{1}{2})| + 1 = |3 + 2| + 1 = 6 \] (True)2. For \( x = 2 \): \[ |3 - 4(2)| + 1 = |3 - 8| + 1 = 6 \] (True)Both solutions satisfy the original equation.

Key Concepts

Absolute Value

Understanding absolute value is crucial when solving equations like \(|3 - 4x| + 1 = 6\). Absolute value is a measure of how far a number is from zero, ignoring whether it is positive or negative. In mathematical terms, it is the number's distance from zero on the number line. For any real number \(a\), the absolute value is denoted as \(|a|\), and it is always non-negative. This concept implies that splitting the original equation into two different cases is necessary since the expression inside the absolute value could be either positive or negative.

• Absolute value turns negative numbers into positive.
• The expression \(|a| = b\) has two potential solutions: \(a = b\) or \(a = -b\).

Understanding this concept helps you deconstruct the absolute value equation into regular algebraic equations, facilitating the solution process.

Isolation of the Variable

To solve any equation efficiently, especially one involving absolute value, isolating the absolute value expression is a fundamental first step. For the equation \(|3 - 4x| + 1 = 6\), the goal in this step is to eliminate components around the absolute value to simplify your work. This can be accomplished by performing inverse operations.
Here's how you approach this:

• Subtract 1 from both sides of the equation to effectively move any constant terms away from the absolute value.
• This yields the simpler form \(|3 - 4x| = 5\), removing any extraneous terms.

Only when the absolute value is isolated, can you confidently separate and solve for the variable embedded within the absolute value expression.

Checking Solutions

Once solutions are found for the transformed equations, verifying each one ensures they satisfy the initial absolute value equation. This verification is essential, especially due to the unique properties of absolute values which can sometimes generate extraneous solutions.
To check each solution:

• Substitute each solution back into the original equation.
• Calculate both sides to confirm equality.

For \(x = -\frac{1}{2}\), substituting back confirms: \( |3 - 4(-\frac{1}{2})| + 1 = |3 + 2| + 1 = 6 \). This ensures it is a valid solution.
Similarly, for \(x = 2\), the verification follows: \( |3 - 4(2)| + 1 = |3 - 8| + 1 = 6 \). Again, confirming validity.
Thus, both \(x = -\frac{1}{2}\) and \(x = 2\) satisfy the original equation.

Intermediate Algebra

When dealing with equations involving absolute value, like \(|3 - 4x| + 1 = 6\), having an understanding of intermediate algebra is beneficial. This includes being comfortable with algebraic manipulations such as distributing, combining like terms, and inverse operations.
Intermediate algebra techniques carry over into various algebraic processes:

• Utilizing inverse operations to isolate terms is a starting point in solving equations efficiently.
• Breaking down complex expressions, like absolute value equations, into simpler forms and solving each separately is a key strategy.

Acquiring these skills not only boosts your ability to solve problems involving absolute value but also enhances your overall mathematical problem-solving toolkit.