Problem 22 - Elementary and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The solution is \(x = 66\).

Step 1: Square Both Sides

To eliminate the square root, square both sides of the equation. This results in:\[\left(\sqrt{\frac{1}{2} x + 3}\right)^2 = 6^2\]Simplifying the equation, we get:\[\frac{1}{2} x + 3 = 36\]

Step 2: Isolate the Term with x

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

Step 3: Solve for x

Multiply both sides by 2 to solve for \(x\):\[x = 66\]This gives us the solution for \(x\).

Key Concepts

Squaring Both Sides

When faced with a radical equation, just like the one we have with a square root, the first step to solving it is to remove the square root. Squaring both sides of the equation is a powerful tool to achieve this. Essentially, what you're doing is raising the equation, including both sides, to the power of two.
This process involves enacting a simple rule: if you have \( \sqrt{a} = b \), then squaring both sides results in \( a = b^2 \). For our exercise, we square both sides of

• The left side: \( \left(\sqrt{\frac{1}{2} x + 3}\right)^2 \)
• The right side: \( 6^2 = 36 \)

Once you square the square root, it cancels out the root itself, leaving you with an algebraic expression without the square root. This is crucial as it simplifies the process greatly and allows us to solve for \(x\) much more easily.
Remember to always be careful when squaring both sides, especially if handling negative numbers or more complex expressions, as it can affect the solution set.

Isolating Variables

Once the radical is eliminated, your goal is to isolate the variable—make it stand alone on one side. This process involves a few simple algebraic maneuvers that help pinpoint the value of \(x\).
After squaring both sides, we are left with the equation:\[\frac{1}{2} x + 3 = 36\]To isolate \(x\), you should perform operations that "undo" whatever has been done to \(x\). For instance, subtract the constant term \(3\) (added to \(x\)) from both sides:
- Subtract 3 from both sides: \( \frac{1}{2} x + 3 - 3 = 36 - 3 \) This simplifies to:- \( \frac{1}{2} x = 33 \) By isolating variables in this manner, you step closer to solving for \(x\). The key is doing what is necessary to make \(x\) by itself on one side of the equation. These techniques are foundational for tackling any algebraic problem effectively.

Algebraic Expressions

Algebraic expressions are the building blocks of equations and help in solving problems like the one in this exercise. They consist of numbers, variables, and arithmetic operations.
In our equation, \(\frac{1}{2} x + 3 = 36\), you can see these elements at work.

• The expression \(\frac{1}{2} x + 3\) is what we call an 'algebraic expression'.
• The \(\frac{1}{2} x\) bit is a 'term' of that expression, involving the multiplication of a fraction with the variable \(x\)

When dealing with such expressions, it's vital to understand each component within them. This helps in manipulating and simplifying them effectively. To solve for \(x\), we apply operations to the entire expression.
For instance, multiplying both sides by 2 helps clear the fraction in \(\frac{1}{2} x\), making things simpler:\[\frac{1}{2} x \times 2 = 33 \times 2\]This turns the equation into \(x = 66\). Understanding and working with algebraic expressions allows us to apply logical methods to find the unknown variable, turning the equation into a simple solvable statement.