Problem 100 - Intermediate Algebra

Question

Solve. $$ \left|\frac{4-3 x}{5}\right|=12 $$

Step-by-Step Solution

Answer

The solutions are \( x = -\frac{56}{3} \) and \( x = \frac{64}{3} \).

Step 1: Understand the Absolute Value Equation

The given equation is \( \left|\frac{4-3x}{5}\right| = 12 \). Absolute value equations of the form \( |A| = B \) imply that \( A = B \) or \( A = -B \).

Step 2: Remove the Absolute Value by Setting Up Two Equations

Since \( \left|\frac{4-3x}{5}\right| = 12 \), we can write two equations: \( \frac{4-3x}{5} = 12 \) and \( \frac{4-3x}{5} = -12 \).

Step 3: Solve the First Equation \( \frac{4-3x}{5} = 12 \)

Multiply both sides by 5 to get rid of the fraction: \[ 4 - 3x = 60 \] Subtract 4 from both sides: \[ -3x = 56 \] Divide by -3: \[ x = -\frac{56}{3} \]

Step 4: Solve the Second Equation \( \frac{4-3x}{5} = -12 \)

Multiply both sides by 5: \[ 4 - 3x = -60 \] Subtract 4 from both sides: \[ -3x = -64 \] Divide by -3: \[ x = \frac{64}{3} \]

Step 5: Verify the Solutions

Substitute \( x = -\frac{56}{3} \) back into the original equation: \[ \left|\frac{4 - 3\left(-\frac{56}{3}\right)}{5}\right| = \left|\frac{4 + 56}{5}\right| = \left|\frac{60}{5}\right| = 12 \]Substitute \( x = \frac{64}{3} \) back into the original equation: \[ \left|\frac{4 - 3\left(\frac{64}{3}\right)}{5}\right| = \left|\frac{4 - 64}{5}\right| = \left|\frac{-60}{5}\right| = 12 \]Both values satisfy the original equation.

Key Concepts

Solving Absolute Value Equations

To solve absolute value equations, we are dealing with equations like \( |A| = B \), where \( A \) is some expression. The absolute value, indicated by the vertical bars, represents the distance of a number from zero. Therefore, it is always non-negative.
When you have an absolute value equation, you need to understand that it implies two separate equations. The expression inside the absolute value can either be equal to \( B \) or negative \( B \).

• The equation \( |A| = B \) translates to \( A = B \) or \( A = -B \).
• This is critical because it recognizes the two possible scenarios that satisfy the equation.

For example, in our exercise problem \( \left|\frac{4-3x}{5}\right|=12 \), we identify two potential equations by considering both possible values for \( A \).

Removing Absolute Value

Removing absolute values from an equation is akin to unraveling a hidden truth. By setting up two equations, we break down the problem to a simpler form. When you write \( |A| = B \), think of it like shedding a layer of abstraction.

• Start by splitting into \( A = B \) and \( A = -B \).
• Ensure that both forms consider the entire expression inside the absolute value, including any fractions or coefficients.

This process is essential because it reveals two possible pathways to solving for the variable. For the given equation \( \left|\frac{4-3x}{5}\right|=12 \), remove the absolute value by considering \( \frac{4-3x}{5}=12 \) and \( \frac{4-3x}{5}=-12 \).
This strategy effectively gives you two simpler linear equations to solve.

Two-Equation Method

The two-equation method is a strategic technique used when dealing with absolute value equations. This method allows us to handle the potential positive and negative cases separately, ensuring no possible solution is overlooked.

• After removing the absolute value, you'll solve each equation individually.
• In our exercise, this approach simplifies to two linear equations.

Take the first equation \( \frac{4-3x}{5} = 12 \), multiply through by 5 to eliminate the fraction, and solve for \( x \). Do the same for \( \frac{4-3x}{5} = -12 \).
Once you find solutions for both equations, you have a complete set of potential solutions for the original absolute value equation.

Solution Verification

After solving your equations, don't forget to verify your solutions! This step is crucial to confirm that both values satisfy the original absolute value equation. Verification acts as a check to avoid errors and ensure correctness.
Substitute each solution back into the original equation one at a time.

• Calculate and confirm the absolute value equals the number on the right side.
• For this exercise, check both \( x = -\frac{56}{3} \) and \( x = \frac{64}{3} \).

With verification, you establish that each proposed solution is valid, meeting the equation's requirements without discrepancies.
Having this assurance allows you to confidently claim your solutions are correct and accurate.