Problem 76 - Algebra and Trigonometry

Question

In Exercises 61–78, solve each absolute value equation or indicate that the equation has no solution. $$|3 x-2|+4=4$$

Step-by-Step Solution

Answer

The solution to \(|3x - 2| + 4 = 4\) equation is \(x = 2/3\).

Step 1: Isolate the absolute value term

In order to find the value of 'x' that satisfies the equation, the term with the absolute value needs to be isolated. In this case, subtract '4' from both sides of the equation to isolate \(|3x - 2|\). This gives us the equation \(|3x - 2| = 0\).

Step 2: Solve the absolute value equation

The absolute value of a number equals zero only when the number itself is zero. To solve the equation \(|3x - 2| = 0\), set the expression inside the absolute value sign equal to zero. This will set up the equation \(3x - 2 = 0\). Now solve this equation for 'x'.

Step 3: Solve for x

To solve for 'x', add '2' to both sides of the equation \(3x - 2 = 0\), giving us \(3x = 2\). Now divide both sides of this equation by '3' to isolate 'x'. This gives us \(x = 2/3\).

Key Concepts

Solving Absolute Value Equations

Understanding how to solve absolute value equations is essential when dealing with real-life problems related to distance and magnitude. These equations include an absolute value, which represents the distance of a number from zero on a number line, without considering the direction.

To solve an absolute value equation, like the given exercise \( |3x - 2| + 4 = 4 \), the first step is recognizing the need to remove any additional terms that do not include the absolute value. This simplification step is crucial for isolating the absolute value expression, allowing us to tackle the core of the equation directly.

Once isolated, we can interpret the absolute value equation as two separate linear equations — one for the positive scenario and another for the negative. However, in the given exercise, since the absolute value equals zero after isolation, there's only one solution, as the absolute value expression must itself be zero.

Isolate the Absolute Value Term

The key to solving equations with absolute values is to first isolate the absolute value term. This creates a clear path to identify the solutions. Isolation means getting the absolute value term alone on one side of the equation.

For the given exercise, starting with \( |3x - 2| + 4 = 4 \), we must eliminate any constants or other terms that are not within the absolute value. By subtracting '4' from both sides of the equation, as shown in the solution, we successfully isolate the absolute value term: \( |3x - 2| = 0 \).

Isolating the absolute value is a methodical process that sets the stage for more straightforward calculations, enabling us to focus solely on solving for the variable contained within the absolute value.

Algebraic Steps to Solve Equations

Once we have isolated the absolute value term, we use algebraic steps to solve the equation. Algebraic manipulation involves applying arithmetic operations systematically to both sides of an equation to maintain equality and solve for the unknown variable.

In our exercise, after isolation, we set the inside of the absolute value term equal to zero: \( 3x - 2 = 0 \). To solve for 'x', the next step is adding '2' to both sides, giving us \( 3x = 2 \). Finally, we divide both sides by '3' to find the value of 'x', which is \( x = \frac{2}{3} \).

Following this sequence—performing inverse operations and simplifying—leads us to the solution. It's critical for students to understand these algebraic steps and the underlying logic to become proficient in solving a wide range of algebraic problems.