Problem 29 - College Algebra

Question

For the following exercises, solve the equation involving absolute value. $$ |3 x-4|=8 $$

Step-by-Step Solution

Answer

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

Step 1: Understand the Definition of Absolute Value

The absolute value of a number represents its distance from zero on the number line, without considering direction. Thus, the equation \(|3x - 4| = 8\) means that the expression inside the absolute value, \(3x - 4\), can be equal to 8 or \(-8\).

Step 2: Set Up Two Separate Equations

Since \(|3x - 4| = 8\), we can write two separate equations: (1) \(3x - 4 = 8\) and (2) \(3x - 4 = -8\). These represent the two possible values of \(3x - 4\) that will satisfy the absolute value equation.

Step 3: Solve the First Equation

Start with the first equation \(3x - 4 = 8\). Add 4 to both sides to isolate the term with \(x\): \[3x = 12\] Next, divide both sides by 3 to solve for \(x\): \[x = 4\]

Step 4: Solve the Second Equation

Now solve the second equation \(3x - 4 = -8\). Add 4 to both sides to isolate the term with \(x\): \[3x = -4\] Next, divide both sides by 3 to solve for \(x\): \[x = -\frac{4}{3}\]

Step 5: State the Solutions

The solutions to the original equation \(|3x - 4| = 8\) are the values of \(x\) found in the previous steps. Thus, \(x = 4\) and \(x = -\frac{4}{3}\) are the solutions.

Key Concepts

Solving Equations

When solving equations, particularly those involving absolute values, it's crucial to grasp the fundamental properties of absolute values. Absolute value is a measure of how far a number is from zero on the number line, disregarding direction.

In absolute value equations, such as \(|3x - 4| = 8\), the solution entails setting up two separate equations.

• The first equation is derived from removing the absolute value brackets and setting the expression equal to 8: \(3x - 4 = 8\).
• The second equation is considering the expression equal to the negative value: \(3x - 4 = -8\).

This stems from the property that the absolute value of a number \(a\) is \(a\) if \(a\) is positive or zero, and \(-a\) if \(a\) is negative. By solving these equations separately, you can find all possible solutions for \(x\) that satisfy the original equation.

Step-by-Step Solution

We can solve the absolute value equation \(|3x - 4| = 8\) by following a systematic approach. This step-by-step process ensures clarity and accuracy.

• Step 1: Set up the Equations. The equation \(|3x - 4| = 8\) translates into two distinct equations: \(3x - 4 = 8\) and \(3x - 4 = -8\).
• Step 2: Solve the First Equation. Tackle \(3x - 4 = 8\). To isolate \(x\), add 4 to both sides obtaining \(3x = 12\). Then, divide by 3 to find \(x = 4\).
• Step 3: Solve the Second Equation. Similarly, solve \(3x - 4 = -8\). Add 4 to both sides: \(3x = -4\), and divide by 3 to get \(x = -\frac{4}{3}\).
• Step 4: Conclusion. The solutions are \(x = 4\) and \(x = -\frac{4}{3}\).

Understanding each step's role helps solidify the process of solving absolute value equations, ensuring you can handle similar problems independently.

Distance on Number Line

The concept of distance on a number line is foundational in understanding absolute value. Distance, in this context, refers to how far one point is from another, specifically from zero, without considering direction.

Absolute value expresses this distance. For instance, if you visualize the line, \(|3x - 4| = 8\) addresses the distance from the value of \(3x - 4\) to zero, which equals 8. Thus, the expression inside the absolute value can be either 8 or -8.

Visualizing this concept may involve laying out the number line and marking points for 8 and -8. This visualization can help intuitively understand why there are two parts to solving absolute value equations. Emphasizing distance without direction makes absolute values distinct and introduces unique characteristics to solving these equations.