Problem 10 - Intermediate Algebra

Question

Solve. \(\sqrt{3 x+3}-4=8\)

Step-by-Step Solution

Answer

The solution is \(x = 47\).

Step 1: Isolate the square root

Let's move the constant term to the other side of the equation. We do this by adding 4 to both sides:\[\sqrt{3x + 3} - 4 + 4 = 8 + 4\]Which simplifies to:\[\sqrt{3x + 3} = 12\]

Step 2: Eliminate the square root

We can eliminate the square root by squaring both sides of the equation. This means raising both sides to the power of 2:\[(\sqrt{3x + 3})^2 = 12^2\]This simplifies to:\[3x + 3 = 144\]

Step 3: Solve for x

Now, we solve the resulting equation. Start by subtracting 3 from both sides:\[3x + 3 - 3 = 144 - 3\]Which simplifies to:\[3x = 141\]Next, divide both sides by 3 to find the value of \(x\):\[x = \frac{141}{3}\]This simplifies to:\[x = 47\]

Key Concepts

Isolating the Square Root

Before you can efficiently solve a radical equation like \( \sqrt{3x + 3} - 4 = 8 \), the first vital step is isolating the square root. In this process, our goal is to get the square root term all by itself on one side of the equation. This allows us to work with the expression inside the square root more easily.
Consider your current equation: you want to isolate \( \sqrt{3x + 3} \). To do this, you will need to add 4 to both sides of the equation to eliminate the -4 that is on the left side.

After performing the operation, the equation will simplify to \( \sqrt{3x + 3} = 12 \). Using this strategy helps in preparing the equation for the next steps. In summary:

• Identify the square root term that needs isolating.
• Perform necessary arithmetic actions (like adding or subtracting) to get it alone.
• This creates a simpler equation to handle for the next step of solving.

Eliminating the Square Root

Once the square root is isolated, the next step in solving the equation is to eliminate the square root itself. This is done by applying a simple operation: squaring both sides of the equation. Remember, the main focus here is to remove the square root by performing the inverse operation, which is squaring.
Consider the square root equation from the earlier step: \( \sqrt{3x + 3} = 12 \). When you square both sides, you are essentially raising each to the power of 2:

• The left side becomes: \( (\sqrt{3x + 3})^2 = 3x + 3 \).
• The right side: \( 12^2 = 144 \).

This gives us a radical-free equation, \( 3x + 3 = 144 \), which is much simpler to solve. A tip to remember: squaring both sides removes the radical but make sure no extraneous solutions are introduced since squaring can sometimes result in additional, irrelevant solutions.

Solving for x

With the square root eliminated, the equation \( 3x + 3 = 144 \) is now an algebraic equation ready for basic solving techniques to find \( x \). This is where your algebra skills come into play. Let’s break it down in a way that’s easy to follow.
First, isolate the \( x \)-term by subtracting 3 from both sides:

• Subtract 3: \( 3x + 3 - 3 = 144 - 3 \).
• This simplifies to: \( 3x = 141 \).

Next, to solve for \( x \), divide both sides by 3:

• Dividing gives: \( x = \frac{141}{3} \).
• Now, simplify the fraction to find \( x = 47 \).

Understanding these steps ensures you can solve for \( x \) comfortably. Solving equations involves basic operations, often reversing a series of steps, which in this case led us to \( x = 47 \), the final solution.