Problem 30 - College Algebra with Corequisite Support

Question

For the following exercises, solve the equation involving absolute value. $$ |2 x-3|=-2 $$

Step-by-Step Solution

Answer

The equation \(|2x - 3| = -2\) has no solutions.

Step 1: Understand the Absolute Value Concept

The equation contains an absolute value term, \(|2x - 3|\). Absolute values represent the distance from zero, meaning they are always non-negative.

Step 2: Analyze the Equation

Given equation is \(|2x - 3| = -2\). However, an absolute value expression cannot equal a negative number because distance (absolute value) cannot be negative.

Step 3: Conclude No Solution

Since \(|2x - 3| = -2\) has no valid solutions in the real number system, because an absolute value cannot equal a negative value, we conclude that the equation has no solutions.

Key Concepts

Algebra

Algebra is the branch of mathematics that uses symbols and letters to represent numbers and quantities in formulas and equations. These symbols can stand in for unknown values, which are often called variables. In algebra, solving equations involves finding values for these variables that make the equation true. When dealing with equations that include absolute values, understanding the basic properties of absolute values is crucial. Absolute values describe the distance a number is from zero on the number line. Since distance cannot be negative, absolute value expressions always yield non-negative results. This principle is foundational when analyzing and solving absolute value equations like \(|2x - 3| = -2\). By applying algebraic techniques, we determine if solutions are possible, which in this case, due to a negative result, there are none.

Real Numbers

Real numbers are a set of numbers that include all the numbers on the number line. This set comprises all the integers, fractions, and irrational numbers like \(\sqrt{2}\) or \(\pi\). When working with real numbers, it's essential to remember that they can be positive, negative, or zero. The exercise involves understanding that no real number can produce a negative absolute value result. Consequently, when the equation \(|2x - 3| = -2\) is given, it's impossible to identify a real number \(x\) that satisfies this equation. Simply put, real numbers cannot fulfill a condition where an absolute value equals a negative, so we categorize the problem as having no solutions.

Non-negative

Non-negative numbers are those that are either zero or positive. They play a key role in absolute value equations, where we're dealing with terms representing distances or lengths. Because distance or length can't be less than zero, absolute values are always non-negative. This principle is the cornerstone when solving equations involving absolute values.In the exercise, the equation \(|2x - 3| = -2\) suggests an outcome that defies the very nature of absolute values. The requirement for a result to be non-negative means that no solution exists if we are asked to equate it with a negative number. Recognizing and remembering that absolute values won't produce negative results can simplify many algebraic problems, helping to determine when certain equations have no solutions.