Problem 76 - College Algebra

Question

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 the equation is \(x = 2/3\).

Step 1: Simplify the equation

The equation is \( |3x - 2| + 4 - 4 \). After subtracting 4 from both sides, we get the equation to be \( |3x - 2| = 0 \)

Step 2: Solve for Absolute Value Equal to Zero

The absolute value of a real number equals zero if and only if that number is zero. Thus, \(3x - 2 = 0 \)

Step 3: Solve for x

To find the value of x, we should isolate x on one side of the equation. This is done by firstly adding 2 to both sides, giving \(3x = 2\). Then, we divide both sides by 3, giving the solution as \(x = 2/3\)

Key Concepts

Solving Absolute Value Equations

When we solve absolute value equations, our goal is to find the values of the variable that satisfy the equation. Absolute value refers to the distance of a number from zero on the number line, regardless of direction. This means it is always a non-negative value.

So, an absolute value equation can often represent two possible equations. However, in this specific case, the equation got simplified to just one possible equation. Here, the equation, \(|3x - 2| = 0\), means that the expression inside the absolute value must itself be zero.

To solve \(|3x - 2| = 0\), we then remove the absolute value and set the equation inside to zero: \(3x - 2 = 0\). This uniquely determines one specific value for \(x\).

Algebraic Manipulation

Algebraic manipulation entails using mathematical operations and properties to conveniently re-arrange and simplify expressions and equations. In the case of solving \(3x - 2 = 0\), we utilized simple manipulation steps.

• First, we sought to isolate \(3x\) by adding 2 to both sides of the equation, resulting in \(3x = 2\).
• Next, we made \(x\) the subject by dividing both sides of the resultant equation by 3. This gave us \(x = \frac{2}{3}\).

These manipulations were straightforward and are based on fundamental algebraic principles. Always perform the same operation on both sides of the equation to maintain equality and gradually solve for the variable.

Role of Real Numbers in Equations

Real numbers are the foundation of mathematical equations. They include all the numbers on the number line, encompassing both rational numbers like fractions and integers, as well as irrational numbers.

In the exercise, the real numbers involved are the result of solving the expression \(3x - 2 = 0\). Specifically, the solution \(x = \frac{2}{3}\) is a real number. Real numbers are crucial in algebra because they allow for a broad and smooth range of solutions, making them essential for solving equations.

When dealing with absolute value equations, understanding the set of real numbers helps us gauge possible solutions, especially since absolute values return real, non-negative results. This expected characteristic of real numbers plays a pivotal role in solving and confirming the solutions' validity.