Problem 5 - The Calculus with Analytic Geometry

Question

In Exercises 1 through 10, solve for \(x\). $$ |5 x-3|=|3 x+5| $$

Step-by-Step Solution

Answer

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

Step 1: Understand the absolute value equation

Recall that for any two expressions with absolute values, \( |A| = |B| \) implies \( A = B \) or \( A = -B \). Here, we have \( |5x - 3| = |3x + 5| \)

Step 2: Setup first case

For the first case, set the expressions inside the absolute value equal to each other: \[ 5x - 3 = 3x + 5 \]

Step 3: Solve the first case

Solve for \(x\) in the first case: \[ 5x - 3 = 3x + 5 \] Subtract \(3x \) from both sides: \[ 2x - 3 = 5 \] Add 3 to both sides: \[ 2x = 8 \] Divide by 2: \[ x = 4 \]

Step 4: Setup second case

For the second case, set one expression equal to the negative of the other: \[ 5x - 3 = -(3x + 5) \]

Step 5: Solve the second case

Solve for \(x\) in the second case: \[ 5x - 3 = -3x - 5 \] Add \(3x \) to both sides: \[ 8x - 3 = -5 \] Add 3 to both sides: \[ 8x = -2 \] Divide by 8: \[ x = -\frac{1}{4} \]

Step 6: State the solutions

Combine the results from both cases: The solutions to the equation are \( x = 4 \) and \( x = -\frac{1}{4} \).

Key Concepts

solving absolute value equations

Solving absolute value equations involves understanding the basic property of absolute values: \( |A| = |B| \) implies \( A = B \) or \( A = -B \). This is because the absolute value of a number represents its distance from zero, making both the positive and negative versions of the number valid solutions. For instance, in our equation \( |5x - 3| = |3x + 5| \), we consider both possibilities. Remember, absolute values measure magnitude without regard to direction. Thus, always set up both possible cases to cover all potential solutions.

case analysis in equations

Case analysis is a critical step when solving absolute value equations. Start by setting the expressions inside the absolute values equal to each other. This is the first case: \[ 5x - 3 = 3x + 5 \]. Solve it step-by-step to find one potential solution.
If two values are equal in magnitude, yet their signs are opposite, you'll need the second case: \[ 5x - 3 = -(3x + 5) \]. Here, you set one expression equal to the negative of the other. This covers the scenario where one side is the negative counterpart of the other.

algebraic manipulation

Algebraic manipulation is the process of rearranging equations and expressions to isolate the variable you are solving for. Follow these steps:

• Start with the equation: \[ 5x - 3 = 3x + 5 \]
• Subtract \(3x\) from both sides: \[ 5x - 3 - 3x = 3x + 5 - 3x \], simplifying to \[ 2x - 3 = 5 \]
• Add 3 to both sides: \[ 2x - 3 + 3 = 5 + 3 \], simplifying to \[ 2x = 8 \]
• Divide by 2: \[ x = 4 \]

For the second case: \[ 5x - 3 = -(3x + 5) \], follow similar steps:

• Expand the negative sign: \[ 5x - 3 = -3x - 5 \]
• Add \(3x\) to both sides: \[ 5x + 3x - 3 = -3x + 3x - 5 \], simplifying to \[ 8x - 3 = -5 \]
• Add 3 to both sides: \[ 8x - 3 + 3 = -5 + 3 \], simplifying to \[ 8x = -2 \]
• Divide by 8: \[ x = -\frac{1}{4} \]

Combining both results gives us the solutions: \( x = 4 \) and \( x = -\frac{1}{4} \).