Problem 22 - Intermediate Algebra

Question

Solve each equation. See Example 1. $$ \sqrt{\frac{1}{2} x+3}=6 $$

Step-by-Step Solution

Answer

The solution to the equation is \( x = 66 \).

Step 1: Understand the Equation

The equation \( \sqrt{\frac{1}{2} x + 3} = 6 \) has a square root on the left-hand side. We need to remove the square root to solve for \( x \).

Step 2: Square Both Sides

Square both sides of the equation to eliminate the square root. \[(\sqrt{\frac{1}{2} x + 3})^2 = 6^2\]This simplifies to:\[\frac{1}{2} x + 3 = 36\]

Step 3: Isolate the Variable Term

Subtract 3 from both sides to isolate the term containing \( x \):\[\frac{1}{2} x = 36 - 3\]This simplifies to:\[\frac{1}{2} x = 33\]

Step 4: Solve for \( x \)

Multiply both sides by 2 to solve for \( x \):\[x = 33 \times 2\]This simplifies to:\[x = 66\]

Step 5: Verify the Solution

Substitute \( x = 66 \) back into the original equation to verify:\[\sqrt{\frac{1}{2} \times 66 + 3} = \sqrt{33 + 3} = \sqrt{36} = 6\]Since both sides of the equation are equal, \( x = 66 \) is the correct solution.

Key Concepts

Understanding Square Roots in Equations

Square roots are numbers that, when multiplied by themselves, yield the original number under the root. In algebra, square roots often appear in equations that we need to solve. Let's consider an equation with a square root, such as \[ \sqrt{\frac{1}{2} x + 3} = 6 \].To solve such equations:

• Identify the entire expression under the square root.
• Square both sides of the equation to remove the square root. This step leaves you with a simpler equation without the square root, which is often easier to solve.

By squaring both sides, \( (\sqrt{\frac{1}{2} x + 3})^2 = 6^2 \), it simplifies to \( \frac{1}{2} x + 3 = 36 \). This technique is crucial, as it helps retain the equality by removing the square root and thus simplifies further manipulations in solving the equation.

Mastering Algebraic Manipulation

Algebraic manipulation is used to simplify equations and solve for the unknown variables effectively. Once you have an equation like\[ \frac{1}{2} x + 3 = 36 \],you need to isolate the variable, here represented as \( x \). Follow these steps:

• Subtract constants from both sides: This means taking away 3 from both sides, giving \( \frac{1}{2} x = 33 \).
• Clear fractions by multiplying both sides by the denominator. Since \( \frac{1}{2} \) is the coefficient, multiply by 2 to get \( x = 66 \).

This results in a clean solution for the variable. Understanding how to perform such algebraic steps allows you to handle more complex equations with ease.

Verification of the Solution

Verifying the solution is a key step to ensure that the found solution truly satisfies the original equation. Begin by substituting the value of \( x \) back into the original equation. For our case, substitute \[ x = 66 \] back:

• Calculate: \( \sqrt{\frac{1}{2} \times 66 + 3} = \sqrt{33 + 3} = \sqrt{36} = 6 \). This verifies that the original equation holds true when \( x = 66 \).
• Check for any extraneous solutions, especially those that might arise from squaring both sides, but here it's straightforward because the comparison is valid.

Verification helps affirm that no steps were skipped or assumed incorrectly during the process and is a validation of both our solution and our understanding of the mathematical procedures involved.