Problem 52 - Calculus with Concepts in Calculus

Question

Solve the equation. $$ |3-4 x|=2 $$

Step-by-Step Solution

Answer

The solutions are \( x = \frac{1}{4} \) and \( x = \frac{5}{4} \).

Step 1: Understand the Absolute Value Equation

The equation given is \[ |3 - 4x| = 2 \]This absolute value equation represents the distance from zero, meaning that the expression inside the absolute value can be both 2 and -2.

Step 2: Set Up the Positive Case

For the positive case, set the expression inside the absolute value equal to 2:\[ 3 - 4x = 2 \]

Step 3: Solve for x in Positive Case

Subtract 3 from both sides:\[ -4x = 2 - 3 \]\[ -4x = -1 \]Now, divide both sides by -4 to solve for \( x \):\[ x = \frac{-1}{-4} \]\[ x = \frac{1}{4} \]

Step 4: Set Up the Negative Case

For the negative case, set the expression inside the absolute value equal to -2:\[ 3 - 4x = -2 \]

Step 5: Solve for x in Negative Case

Subtract 3 from both sides:\[ -4x = -2 - 3 \]\[ -4x = -5 \]Now, divide both sides by -4 to solve for \( x \):\[ x = \frac{-5}{-4} \]\[ x = \frac{5}{4} \]

Step 6: Combine Solutions

Combine the solutions from both cases. The solutions to the equation \(|3-4x| = 2\) are \[ x = \frac{1}{4} \] and \[ x = \frac{5}{4} \].

Key Concepts

Solving Absolute Value Equations

When solving absolute value equations, we're essentially trying to find solutions for an equation that involves the absolute value of a variable expression. The absolute value itself refers to the distance a number is from zero on a number line, regardless of direction. In other words, the absolute value of both positive and negative numbers will become positive. Consider an equation like

• \[ |3-4x|=2 \]

Here, the expression inside the absolute value (\(3-4x\)) can equate to both positive and negative counterparts of 2. So, you set up and solve two separate equations, \(3-4x = 2\) and \(3-4x = -2\). Both of these cases must be solved to find the potential solutions for \(x\).
By solving both cases, you account for the definition of absolute value as a measure of distance in either direction from zero.

Distance From Zero

The concept of absolute value being the distance from zero plays a crucial role when working with equations involving absolute values. Distance from zero is a way to interpret absolute values in mathematical terms. Simply put, numbers like 2 and -2 are both 2 units away from zero, hence their absolute values are both 2.
In the equation \[ |3-4x| = 2 \], the expression inside the absolute value can be seen as traveling 2 units from zero, either in the positive direction, becoming 2, or reversing direction becoming -2. This understanding will allow you to set up two different cases, accounting for both possibilities of movement on the number line, leading to the solutions that satisfy the equation.

Positive and Negative Cases

To thoroughly tackle an absolute value equation, it's essential to investigate both positive and negative cases. This process ensures that all potential solutions are considered. Begin by looking at the inside of the absolute value symbol: \(3-4x\).

• For the positive case, equate it to 2: \(3 - 4x = 2\).
• For the negative case, equate it to -2: \(3 - 4x = -2\).

After setting up these two scenarios, proceed to solve each one separately. The distinct cases handle the two-sided nature of absolute values: moving both to the right of zero and to the left.
This dual approach inherently respects how absolute value captures distances in both directions on a number line.

Algebraic Manipulation

After setting up the positive and negative cases of an absolute value equation, you'll need algebraic manipulation to isolate the variable. Whether it's \(3-4x=2\) or \(3-4x=-2\), start by straightforward steps:

• First, subtract 3 from both sides:
  For the positive case, you get \(-4x = -1\).
• Next, divide all terms by \(-4\) to solve for \(x\):
  \(x = \frac{1}{4}\).
• Repeat similar steps for the negative case:
  \(-4x = -5\), then \(x = \frac{5}{4}\).

These steps highlight the importance of performing operations to both sides of the equation to transform and simplify, reaching the correct values for \(x\). Once you've done this for both cases, you'll have found all solutions to the original equation, demonstrating how algebra empowers you to solve equations step-by-step.