Problem 34 - College Algebra with Corequisite Support

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.

Step 1: Isolate the Absolute Value Expression

To solve the equation \(|2x + 1| - 2 = -3\), first add 2 to both sides to isolate the absolute value expression. This gives us:\[ |2x + 1| = -1 \].

Step 2: Analyze the Absolute Value

The absolute value of any expression is always non-negative, i.e., \(|expression| \geq 0\). Therefore, an equation \(|expression| = -1\) is not possible, because -1 is a negative number.

Step 3: Conclusion

Since the absolute value cannot be negative, there are no such values of \(x\) that satisfy the equation \(|2x + 1| = -1\). Hence, the equation has no solutions.

Key Concepts

Solving Equations

Solving equations, especially when they involve absolute values, requires careful analysis to ensure all terms are correctly handled. Here's how we approach it:

• Start by identifying the absolute value expression and isolate it, if necessary, by performing operations on both sides of the equation.
• Analyze the equation once the absolute value expression is on one side of the equation completely.

In our case, we began with the equation \(|2x + 1| - 2 = -3\). To isolate |2x + 1|, we added 2 to both sides, resulting in |2x + 1| = -1. This step was essential to simplify the equation before proceeding further.

By isolating the absolute value, you simplify the problem which makes it easier to identify the possible solutions, or in some cases, the lack thereof.

Algebraic Expressions

Understanding algebraic expressions is a cornerstone of solving equations, including those involving absolute values. An algebraic expression like \(2x + 1\) within an absolute value involves operations that need careful consideration when solving the equation.

• Algebraic expressions can be manipulated using basic arithmetic operations such as addition, subtraction, multiplication, and division.
• When part of an absolute value notation, these expressions must first be isolated to simplify the equation-solving process.

The expression \(2x + 1\) within the absolute value brackets indicates a range of possible values for \(x\), but when set equal to a negative number like -1, it presents a conflict since absolute values cannot be negative.

Hence, understanding the structure of algebraic expressions aids in determining potential solutions and identifying contradictions or impossibilities within equations.

Non-negative Values

A fundamental property of absolute values is that they are always non-negative. This means that any expression within an absolute value bracket, such as \(|expression|\), will produce a result that is zero or positive.

• Since absolute values measure distance, they cannot be negative.
• Equations like |2x + 1| = -1 immediately indicate that no solutions exist because -1 is negative.

Recognizing this property early on in solving absolute value equations saves time and prevents attempts to find solutions where none exist.

When you encounter a scenario where an absolute value equals a negative number, it signifies that the equation is unsolvable under real number standards. This demonstrates the importance of thoroughly understanding mathematical properties when solving equations.