Problem 92 - Intermediate Algebra

Question

Find the error in each solution and correct. \(\sqrt{2 x+3}+4=1\) \(\begin{aligned} \sqrt{2 x+3} &=5 \\\\(\sqrt{2 x+3})^{2} &=5^{2} \\ 2 x+3 &=25 \\ 2 x &=22 \\ x &=11 \end{aligned}\)

Step-by-Step Solution

Answer

The error is in assuming \(\sqrt{2x+3} = 5\); the equation \(\sqrt{2x+3} = -3\) has no real solutions.

Step 1: Understanding the Error

The given solution is trying to solve the equation \(\sqrt{2x+3} + 4 = 1\). The mistake is in the very first step where it's assumed that \(\sqrt{2x+3} = 5\). This assumption does not correctly isolate the square root term, leading to an incorrect solution process.

Step 2: Rearranging the Equation

To solve the equation \(\sqrt{2x+3} + 4 = 1\) correctly, subtract 4 from both sides to isolate the square root term: \(\sqrt{2x+3} = 1 - 4\). Simplifying gives \(\sqrt{2x+3} = -3\).

Step 3: Analyzing the Result

A square root of a real number cannot be negative. Therefore, \(\sqrt{2x+3} = -3\) is not possible in the set of real numbers. This indicates that the equation has no solution for real \(x\).

Key Concepts

Equation Solving

Equation solving is a fundamental skill in algebra that involves finding the value of the variable that makes the equation true. It typically starts by simplifying and rearranging the equation. The goal is to isolate the variable on one side of the equals sign. For example, if we have the equation \( \sqrt{2x+3} + 4 = 1 \), we aim to simplify this by performing operations that help us solve for \( x \). This involves using inverse operations to break down the equation step-by-step.

• First, identify operations applied to the variable.
• Apply inverse operations in reverse order: addition is undone by subtraction and multiplication by division.
• Continue simplifying until the variable is isolated.

In our example, subtracting 4 from both sides is key to bringing \( \sqrt{2x+3} \) into an isolated position. However, as explained in the solution, issues arise when further simplifications produce invalid conditions, like a negative square root.

Square Roots

Square roots refer to the inverse operation of squaring a number. The square root of a number \( y \) is a number \( x \) such that \( x^2 = y \). When dealing with square roots in algebra, especially in solving equations, it is crucial to understand their properties. One key fact is that the square root function, when dealing with real numbers, does not yield negative results.

• The notation \( \sqrt{x} \) represents the non-negative root of \( x \).
• If you encounter \( \sqrt{x} = -a \) where \( a > 0 \), then there is no real number solution.

In the provided exercise, attempting to accept a negative value, like \( \sqrt{2x+3} = -3 \), leads to an error. This is because it contradicts the definition of square roots over real numbers, indicating the importance of verifying the feasibility of solutions.

Isolating Terms

Isolating terms is a strategy used in algebra to solve equations by getting the variable of interest alone on one side. This involves manipulating the equation using opposite operations to those initially applied to the terms involving the variable.

• Identify what needs to be isolated. Here, it is \( \sqrt{2x+3} \).
• Use inverse operations: if a term is added, subtract it; if multiplied, divide it.

In the original problem, the equation \( \sqrt{2x+3} + 4 = 1 \) was incorrectly handled by not first isolating \( \sqrt{2x+3} \) properly. The correct approach was to subtract 4 from both sides, leading to \( \sqrt{2x+3} = -3 \). Since this outcome isn't valid for real numbers, it highlights the need for careful isolation and verification of the term, ensuring it does not violate underlying mathematical principles. This process of step-by-step isolation helps identify when an equation has no solution.