Problem 34 - Intermediate Algebra

Question

Solve each equation. See Example 2. $$ |6 x-3|+7=28 $$

Step-by-Step Solution

Answer

The solutions are \(x = 4\) and \(x = -3\).

Step 1: Isolate the Absolute Value

Start by isolating the absolute value expression. To do this, subtract 7 from both sides of the equation: \[|6x - 3| + 7 - 7 = 28 - 7\]which simplifies to:\[|6x - 3| = 21.\]

Step 2: Set Up Two Equations

The absolute value expression \(|6x - 3| = 21\) means that \(6x - 3\) can be either 21 or -21. Therefore, we set up two separate equations:Equation 1: \[6x - 3 = 21\]Equation 2: \[6x - 3 = -21.\]

Step 3: Solve Equation 1

Solve the first equation, \(6x - 3 = 21\). Add 3 to both sides:\[6x - 3 + 3 = 21 + 3\]which simplifies to:\[6x = 24.\]Now, divide by 6:\[x = \frac{24}{6}\]which simplifies to:\[x = 4.\]

Step 4: Solve Equation 2

Solve the second equation, \(6x - 3 = -21\). Add 3 to both sides:\[6x - 3 + 3 = -21 + 3\]which simplifies to:\[6x = -18.\]Now, divide by 6:\[x = \frac{-18}{6}\]which simplifies to:\[x = -3.\]

Step 5: Final Answer

The solutions to the original equation \(|6x - 3| + 7 = 28\) are \(x = 4\) and \(x = -3\). These are the values that satisfy the equation.

Key Concepts

Isolation of Terms

The first important step when solving equations involving absolute values is to isolate the absolute value term. This gives clarity and makes it easier to handle the components of the equation. In our exercise, we had the expression \(|6x - 3| + 7 = 28\), which contains an absolute value expression \(|6x - 3|\).

To isolate this, we subtract 7 from both sides of the equation. This process helps in simplifying the equation by reducing additional terms that don't involve the absolute value.
Here’s how it's done step-by-step:

• Start with \(|6x - 3| + 7 = 28\).
• Subtract 7 from both sides to eliminate the constant outside the absolute value: \(|6x - 3| + 7 - 7 = 28 - 7\).
• This simplifies to: \(|6x - 3| = 21\).

By completing this operation, you prepare the equation for further solving by focusing on the isolated absolute value expression.

Equation Solving

Once you've isolated the absolute value, the next phase is solving the equation that emerges from it. Because absolute values can represent either positive or negative scenarios, we create two separate equations.

This step ensures all potential values that solve the equation are considered. Here's a breakdown:

• The absolute value \( |6x - 3| = 21\) translates to two equations because both \(6x - 3 = 21\) and \(6x - 3 = -21\) can satisfy the absolute value condition.
• By setting these as separate equations, you prepare to solve each scenario independently.

The key is recognizing the dual nature of absolute values, opening up the pathway to find all possible solutions.

Two Equation Approach

When handling absolute value equations, the two equation approach is crucial. This approach derives from the fundamental nature of absolute values: they measure distance from zero, which can equate to two possible values.

For our problem, we breakdown as follows:

• First equation: \(6x - 3 = 21\). Solve it step-by-step by adding 3 to both sides (\(6x = 24\)), and then dividing by 6 (\(x = 4\)).
• Second equation: \(6x - 3 = -21\). Similarly, add 3 to both sides (\(6x = -18\)), and divide by 6 (\(x = -3\)).

This systematic approach helps ensure that both potential solutions derived from the absolute value are considered, leading to a complete solution set for the original equation. Remember, each equation forms a pathway to a unique solution, emphasizing the importance of solving each possibility diligently.