Problem 65 - College Algebra

Question

Solve each absolute value equation or indicate the equation has no solution. $$ 2|3 x-2|=14 $$

Step-by-Step Solution

Answer

The solutions are \(x = 3\) and \(x = -5/3\).

Step 1: Interpret absolute value

An absolute value equation can be translated into two separate equations. So, here it will branch into: \(3x - 2 = 7\) and \(3x - 2 = -7\)

Step 2: Solve the first equation

Let's solve the first equation; \(3x - 2 = 7\). Adding 2 to both sides, we get \(3x = 9\). Dividing each side by 3, we get \(x = 3\).

Step 3: Solve the second equation

Now, solve the second equation; \(3x - 2 = -7\). Adding 2 to both sides, we get \(3x = -5\). Dividing each side by 3, we get \(x = -5/3\) or -1.67 to two decimal places.

Key Concepts

Solving Absolute Value Equations

In mathematics, absolute values represent the distance a number is from zero on a number line. When tackling absolute value equations, such as \(2|3x-2|=14\), understanding this concept is crucial. The absolute value of an expression affects how equations are set up and solved.

Absolute value equations are solved by splitting the equation into two possible cases. This step arises from the definition of absolute value, which states that \(|a| = a\) if \(a \geq 0\) or \(|a| = -a\) if \(a < 0\). Consequently, the equation \(2|3x-2|=14\) transforms into two potential scenarios to solve:

• \(3x - 2 = 7\)
• \(3x - 2 = -7\)

In each case, we consider both the positive and negative possibilities of the absolute value to ensure we cover both outcomes of the 'distance' aspect inherent in absolute values.

Recognizing the dual scenarios is a key step in accurately solving absolute value equations.

Equation Transformation

Equation transformation is a fundamental process in mathematics, allowing equations to be rewritten in simpler or alternative forms. With absolute value equations, transformation involves setting up equivalent equations to eliminate the absolute value.

In our example, the starting equation is \(2|3x-2|=14\). The first transformation step is to divide both sides by 2 to isolate the absolute value alone:

• \(|3x - 2| = 7\)

Next, this expression translates into two separate linear equations:

• \(3x - 2 = 7\)
• \(3x - 2 = -7\)

This transformation is essential for removing the absolute magnitude constraints and allowing standard algebraic techniques to solve the equation. Equation transformation provides a pathway to simplifying complex equations, making them manageable to solve.

Isolation of Variables

Isolation of variables is an essential algebra technique used to solve for unknowns. When dealing with linear equations, the goal is to manipulate the equation so that the unknown variable stands alone on one side of the equation. This process uncovers the solution by making the variable's value explicit.

In the absolute value equation \(3x - 2 = 7\), the first step is to eliminate constants from the side with the variable. We add 2 to both sides to achieve this:

• \(3x - 2 + 2 = 7 + 2 \rightarrow 3x = 9\)

Next, divide each side by 3 to isolate \(x\):

• \(x = 3\)

The same steps apply to the equation \(3x - 2 = -7\), adding 2 and dividing by 3:

• \(3x = -5 \rightarrow x = -5/3\)

These actions isolate the variable for both branches of the initial absolute value condition.

Algebraic Solutions

The final step in solving equations typically involves obtaining algebraic solutions where the variable of interest is isolated, providing direct answers to the problem.

When simplified, each algebraic equation yields solutions indicating the values for \(x\) that satisfy the original equation. For the first equation, \(3x = 9\), solving gives \(x = 3\). This becomes one valid solution because it maintains the original equation's integrity.

For the second equation, \(3x = -5\), solving offers \(x = -\frac{5}{3}\), another viable solution. Each algebraic solution reflects a possible value the variable may assume to equate both sides of the original absolute value condition.

When assessing absolute value equations, always verify these solutions in the context of the problem to ensure they satisfy the initial constraints. This verification process helps determine the feasibility of the proposed solutions.