Problem 34 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-50: Solve the equation. $$ \sqrt{x+1}=\sqrt{x-1} $$

Step-by-Step Solution

Answer

The equation has no solutions.

Step 1: Understand the Equation

The equation we need to solve is \( \sqrt{x+1} = \sqrt{x-1} \). Here, we have square roots on both sides that we need to simplify.

Step 2: Eliminate the Square Roots

To eliminate the square roots, square both sides of the equation. This gives us \((\sqrt{x+1})^2 = (\sqrt{x-1})^2\). Simplifying this results in the equation: \(x+1 = x-1\).

Step 3: Solve the Simplified Equation

With the equation \(x+1 = x-1\), subtract \(x\) from both sides, resulting in \(1 = -1\). This is a contradiction, indicating that the equation has no solutions.

Step 4: Verify the Result

Since squaring both sides of an equation is not a reversible operation, it's important to check if there could have been solutions missed by the process. However, our simplification led to a contradiction, confirming that no values of \(x\) satisfy the original equation.

Key Concepts

Square Roots

Square roots are mathematical expressions that represent a number which, when multiplied by itself, gives the original number. For instance, the square root of 9 is 3, because 3 times 3 equals 9. Square roots are often denoted using the radical symbol, \( \sqrt{} \). In equations, you might encounter square roots on either side, which can make them seem complex at first. However, they follow some basic rules. For example, the square root of a negative number isn't a real number, but rather an imaginary one. Understanding these rules is crucial when working with square root equations.
To simplify equations involving square roots, we often have to eliminate the square roots by reversing operations, such as squaring both sides of the equation. This helps to transform the radical expressions into a more manageable algebraic form. Again, this process requires careful handling to avoid errors and extraneous solutions.

Eliminating Square Roots

Eliminating square roots from an equation involves removing the radical sign by squaring both sides of the equation. In our example, starting with the equation \(\sqrt{x+1} = \sqrt{x-1}\), we eliminate the square roots by squaring each side:

• \((\sqrt{x+1})^2 = (\sqrt{x-1})^2\)
• This simplifies to \(x+1 = x-1\).

By squaring the square roots, we're able to express the equation in a linear form. This step transforms the equation into one that is often easier to solve. However, it's important to remember that squaring both sides can introduce extra solutions or miss potential restrictions. Therefore, further verification is needed to ensure any resulting solutions are valid.

Contradiction in Equations

A contradiction in an equation occurs when the simplified form of an equation leads to an impossible statement. In our exercise, after squaring both sides of \(\sqrt{x+1} = \sqrt{x-1}\), we obtained \(x+1 = x-1\).

• By simplifying, we found \(1 = -1\), which is obviously false.

This contradiction happens because no number can satisfy this statement. In simple terms, it means that the original equation does not have any solutions. Contradictions signal that either there are no solutions or that something went wrong in the solving process, requiring us to reconsider our steps or assumptions.

Verification of Solutions

Verification of solutions is a crucial step in solving equations. It helps ensure that our solution process didn't introduce errors or unnecessary solutions. When dealing with square roots, remember that squaring is not a reversible action, meaning any perceived solutions after squaring need to be verified against the original equation.
In our example, after finding a contradiction from the elimination step, we realize the contradiction itself proves absence of any solutions. However, when no contradiction occurs, you'd substitute the solutions back into the original equation to confirm their validity. This way, we avoid accepting invalid or extraneous solutions, especially those introduced when both sides of an equation are squared.