Problem 38 - Intermediate Algebra

Question

Solve each equation. See Example 2. $$ \left|\frac{3}{4} x+4\right|-5=11 $$

Step-by-Step Solution

Answer

Solutions are \( x = 16 \) and \( x = -\frac{80}{3} \).

Step 1: Isolate the Absolute Value

To isolate the absolute value, we first need to remove the constant term on the right side of the equation. Add 5 to both sides to get: \( \left| \frac{3}{4}x + 4 \right| = 16 \).

Step 2: Set Up Two Equations

The absolute value equation \(\left| A \right| = B\) translates to two separate equations: \( A = B \) and \( A = -B \). Applying this to our equation, we get: \[ \frac{3}{4}x + 4 = 16 \] and \[ \frac{3}{4}x + 4 = -16 \].

Step 3: Solve the First Equation

Solve \( \frac{3}{4}x + 4 = 16 \) by first subtracting 4 from both sides to find \( \frac{3}{4}x = 12 \). Next, multiply both sides by \( \frac{4}{3} \) to solve for \( x \): \[ x = \frac{4}{3} \times 12 = 16 \].

Step 4: Solve the Second Equation

Solve \( \frac{3}{4}x + 4 = -16 \) by first subtracting 4 from both sides to find \( \frac{3}{4}x = -20 \). Next, multiply both sides by \( \frac{4}{3} \) to solve for \( x \): \[ x = \frac{4}{3} \times (-20) = -\frac{80}{3} \].

Step 5: Conclude the Solution

The solutions to the original equation are \( x = 16 \) and \( x = -\frac{80}{3} \). These solutions satisfy the absolute value equation after the operations performed.

Key Concepts

Isolating the Absolute Value

When working with absolute value equations, one crucial step is to isolate the absolute value expression on one side of the equation. This means you need to get rid of any constants that are affecting the term.
To achieve this in our example, first focus on the equation \( \left| \frac{3}{4} x + 4 \right| - 5 = 11 \). Notice that the constant \(-5\) is present, and we must remove it to simplify the equation.
How do we do that? Simply add 5 to both sides to achieve balance, resulting in:

• \( \left| \frac{3}{4} x + 4 \right| = 16 \)

Now, the absolute value is neatly isolated. This foundational step lets us move forward with clarity when solving the equation.

Solving Equations

After isolating the absolute value, your next task is to address the resulting absolute value equation. You should understand what that equation truly signifies.
In basic terms, the equation \( \left| A \right| = B \) implies that the quantity inside the absolute value, \( A \), can either be equal to \( B \) or its negative counterpart, \(-B\). Why? Because both would result in \( \left| A \right| = B \).
For our example:

• Set up the first equation: \( \frac{3}{4} x + 4 = 16 \)
• Set up the second equation: \( \frac{3}{4} x + 4 = -16 \)

This approach transforms the single absolute value equation into two distinct linear equations. Each must be solved independently to capture all potential solutions.

Two Separate Equations

With two separate equations ready, the process of solving becomes straightforward. Each equation needs to be tackled individually. For the first equation, \( \frac{3}{4} x + 4 = 16 \), follow these steps:

• Subtract 4 from both sides: \( \frac{3}{4} x = 12 \)
• Multiply by the reciprocal, \( \frac{4}{3} \), to solve for \( x \): \( x = 16 \)

For the second equation, \( \frac{3}{4} x + 4 = -16 \):

• Subtract 4 from both sides: \( \frac{3}{4} x = -20 \)
• Again, multiply by the reciprocal, \( \frac{4}{3} \), to find \( x \): \( x = -\frac{80}{3} \)

Together, these solutions \( x = 16 \) and \( x = -\frac{80}{3} \) represent all the possible values for \( x \) in the original absolute value equation. Each solution corresponds to one of the conditions set out by the absolute value property.