Problem 37 - Beginning and Intermediate Algebra

Question

Solve. $$\sqrt{1-2 x}=3$$

Step-by-Step Solution

Answer

The given equation \(\sqrt{1-2x}=3\) has no solution.

Step 1: Write down the given equation

We are given the equation \(\sqrt{1-2x} = 3\).

Step 2: Square both sides of the equation

Since the aim is to remove the square root, we can square both sides of the equation. After squaring both sides, we get: \[(\sqrt{1-2x})^2 = 3^2.\]

Step 3: Simplify the equation

After squaring both sides, we have: \[1 - 2x = 9.\]

Step 4: Solve for x

Now, isolate \(x\) by adding \(2x\) to both sides and subtracting \(9\) from both sides: \[1 - 9 + 2x = 9 + 2x - 9.\] The equation simplifies to: \[-8 + 2x = 2x,\] If we now subtract \(2x\) from both sides: \[-8 = 0.\]

Step 5: Check for a solution

The equation \(-8 = 0\) is a false statement, which means there is no solution for \(x\) that makes the initial equation true. So, the given equation \(\sqrt{1-2x}=3\) has no solution.

Key Concepts

Solving Equations

At the heart of algebra lies the skill of solving equations. An equation is like a balance, and our job is to find the right value for the variable that keeps things balanced on both sides. This task often involves applying operations like addition, subtraction, multiplication, and division to both sides until the variable is isolated.

Consider the initial equation where we had a square root: \( \sqrt{1-2x}=3 \). Here, solving involved removing the square root by squaring both sides. This helps us "undo" the square root, resulting in a solvable algebraic equation. It’s crucial to remember to perform the same operation on both sides to maintain balance. Once squared, this transforms to \( 1-2x = 9 \).

• Identify operations needed to isolate the variable.
• Ensure to do the same to both sides to maintain balance.
• Simplify iteratively while ensuring the equation remains true.

With practice, solving equations becomes more intuitive and you’ll gain confidence tackling more complex ones.

Square Root Equations

Square root equations involve expressions under a square root symbol. One of the most effective ways to solve them is to eliminate the square root by squaring both sides of the equation. This transforms the equation from one form to another that is easier to solve.

For the equation \( \sqrt{1-2x} = 3 \), squaring both sides changed it to an equation without a square root: \( 1 - 2x = 9 \).

• Understand that solving involves increasing both sides by the power of two.
• Be cautious: squaring can introduce extraneous solutions, so verification is key.
• Always interpret results in the context of the original square root problem.

Solving these equations helps build core algebraic skills and enhances understanding of how square roots behave under different operations.

No Solution Cases

Sometimes, after solving an equation, you might end up with a statement that doesn’t make sense, like \( -8 = 0 \). This tells us that there’s no value for the variable that works in the original equation.

Such situations are known as "no solution" cases, and they arise when the operations lead to contradictions. For our equation \( \sqrt{1-2x} = 3 \), the final statement showed a contradiction, informing us there is no number that satisfies the original equation.

• Recognize that not every equation will have a solution.
• Be methodical in solving, looking out for contradictions or impossibilities.
• Double-check the original problem to see if constraints prevent solutions.

Understanding "no solution" scenarios fosters critical thinking and encourages an analytical approach towards algebraic problems.