Problem 34 - College Algebra

Question

For the following exercises, solve the equation involving absolute value. $$ |2 x+1|-2=-3 $$

Step-by-Step Solution

Answer

The equation has no solutions, as absolute values cannot be negative.

Step 1: Isolate the Absolute Value Expression

Starting with the equation \(|2x + 1| - 2 = -3\), we first need to isolate the absolute value expression. This can be done by adding 2 to both sides of the equation, which gives us \(|2x + 1| = -1\).

Step 2: Consider the Nature of Absolute Values

The expression \(|2x + 1| = -1\) suggests that we would be looking for a number whose absolute value is -1. However, this is impossible, as absolute values cannot be negative. Absolute values represent the non-negative distance of a number from zero on the number line.

Step 3: Conclude No Solutions

Since an absolute value cannot be negative, \(|2x + 1| = -1\) has no solution. Therefore, the original equation \(|2x + 1| - 2 = -3\) has no solutions.

Key Concepts

Understanding Algebraic Equations

Algebraic equations are foundational elements in mathematics, representing statements of equality featuring variables and constants. An equation such as \(|2x + 1| - 2 = -3\) contains algebraic expressions that you solve to find the variable's value.
Equations play a crucial role in problem-solving across various fields like physics and engineering. They allow us to describe relationships between quantities and find unknown values.
In algebra, our aim is often to manipulate the equation into a form where the unknown variable, here represented by \(x\), can be directly solved.
To solve the equation, recognizing the structure and components, such as terms, coefficients, and absolute values, is essential. This understanding helps break down complex equations into more manageable steps.

• Look for terms with the variable and constants.
• Identify operations—such as addition or subtraction—applied to them.
• Understand the role of functions like absolute values within the equation.

Building these skills supports more advanced problem-solving and deepens comprehension of how variables interact within algebraic relationships.

Isolation of Absolute Value

When solving equations involving absolute values, isolating the absolute value expression is a critical step. This process involves rearranging the equation so that the absolute value notation stands alone on one side. In the exercise \(|2x + 1| - 2 = -3\), our first task is to isolate \(|2x + 1|\). We achieve this by removing any constants or terms next to the absolute value by performing inverse operations.

In practical terms, this means adding or subtracting numbers from both sides of the equation. For instance, by adding \(2\) to both sides of \(|2x + 1| - 2 = -3\), we obtain \(|2x + 1| = -1\). This isolates the absolute value term, making it clearer what needs to be addressed next.

• Isolate the absolute value by adding or subtracting surrounding terms.
• Aim for the simplest form where \(|expression| = value\).

Isolating the absolute value expression helps to simplify the equation and sets the stage for further analysis to determine potential solutions to the equation.

Exploring Properties of Absolute Value

Absolute value functions are key concepts in algebra. They measure the distance of a number from zero, ensuring the outcome is always non-negative. This property is important because it defines how absolute value equations behave. For instance, if you have \(|2x + 1| = -1\), it immediately suggests a contradiction because absolute values cannot yield negative results.

Therefore, addressing any equation like this involves examining whether the absolute value configuration has a feasible solution. In the problem, we're tasked with finding a solution for \(|2x + 1| - 2 = -3\), which simplifies to \(|2x + 1| = -1\). Given the property that the absolute value is always greater than or equal to zero, the equation is unsolvable.

• Absolute values represent non-negative magnitudes.
• Equations where the absolute value equals a negative number have no solutions.

Understanding the inherent properties of absolute value is fundamental when deciding on the viability of solutions for any algebraic equation involving absolute values.