Problem 67 - Algebra and Trigonometry

Question

In Exercises 61–78, solve each absolute value equation or indicate that the equation has no solution. $$2|3 x-2|=14$$

Step-by-Step Solution

Answer

The solutions to the equation \(2|3x - 2| = 14\) are \(x = 3\) and \(x = -5/3\).

Step 1: Isolate the absolute value

Start by isolating the absolute value on one side of the equation. Divide each side of the equation by 2 to get: \(|3x - 2| = 7\).

Step 2: Create the two possible equations

The definition of absolute value is that \(|a| = a\) if \(a ≥ 0\) and \(|a| = -a\) if \(a < 0\). Therefore, we can create two possible equations from \(|3x - 2| = 7\): (1) when \(3x - 2 = 7\) and (2) when \(3x - 2 = -7\).

Step 3: Solve the two equations

For the first equation: Add 2 to both sides to get \(3x = 9\). Then divide each side by 3 to obtain \(x = 3\). For the second equation: Add 2 to both sides to achieve \(3x = -5\). Then divide each side by 3 to obtain \(x = -5/3\).

Key Concepts

Solving Equations

Solving equations is a fundamental aspect of algebra and it primarily involves finding the value of the unknown variables. When encountering absolute value equations like \( 2|3x - 2| = 14 \), we aim to determine what values of \( x \) satisfy the equation.

• We start by manipulating the equation so that the absolute value expression stands alone on one side. By doing this, we can better analyze the equation and apply algebraic principles.
• Next, since absolute value represents the distance of a number from zero, the equation \( |a| = b \) implies two potential equations: \( a = b \) or \( a = -b \) because the distance should stay the same whether positive or negative.

To solve the given absolute value equation, we first divide both sides by 2. This simplifies the equation to \( |3x - 2| = 7 \). From here, we identify the possible solutions by setting up two separate equations: \( 3x - 2 = 7 \) and \( 3x - 2 = -7 \). Exploring both scenarios helps us gather all possible solutions.

Algebraic Expressions

An algebraic expression is a mathematical statement that includes variables, numbers, and operation signs. In the equation \( |3x - 2| = 7 \), the expression inside the absolute value symbol, \( 3x - 2 \), combines both variables and constants.

• The term "3x" consists of the coefficient 3 and the variable \( x \). This term suggests that "3 times some number (\( x \))" is involved in calculations.
• Meanwhile, "-2" serves as a constant that shifts the entire expression by two units lower.

Understanding these components allows us to isolate parts of the equation step-by-step. It helps clarify how different elements interact, which is essential in solving more complex equations. By dismantling and understanding these pieces, we develop a strong foundation in solving equations effectively.

Isolating the Variable

Isolating the variable means getting the variable on one side of the equation all by itself, helping us see what value will satisfy the equation. This step is key in transforming algebraic expressions into simple equations that we can easily solve.

• Begin by reversing any arithmetic operations that have been applied to the variable. This often involves adding, subtracting, multiplying, or dividing to remove additional numbers on the same side as the variable.
• In our example, to isolate \( x \) for \( 3x - 2 = 7 \), we first add 2 to both sides to counteract the -2. This adjustment grants us \( 3x = 9 \).
• Next, we divide by 3 to solve for \( x \), leading us to \( x = 3 \). Similarly, carry out these isolation steps for the second potential solution \( 3x - 2 = -7 \).

Success in isolating the variable offers us definitive solutions, simplifying our original complex equation into a clear answer. This technique is essential not just for straightforward equations but also as a foundational tool in tackling advanced mathematics.