Problem 10 - Algebra A Combined Function

Question

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

Step-by-Step Solution

Answer

The solution is \( x = 47 \).

Step 1: Isolate the Square Root

The given equation is \( \sqrt{3x+3} - 4 = 8 \). Start by isolating the square root term by adding 4 to both sides of the equation.\[\sqrt{3x+3} = 8 + 4\]\[\sqrt{3x+3} = 12\]

Step 2: Eliminate the Square Root

To eliminate the square root, square both sides of the equation. This removes the square root on the left-hand side.\[(\sqrt{3x+3})^2 = 12^2\]\[3x + 3 = 144\]

Step 3: Solve for x

Now, solve the linear equation by first subtracting 3 from both sides to isolate the term with \( x \).\[3x = 144 - 3\]\[3x = 141\]Next, divide both sides by 3 to solve for \( x \).\[x = \frac{141}{3}\]\[x = 47\]

Step 4: Verify the Solution

To ensure the solution is correct, substitute \( x = 47 \) back into the original equation.\[\sqrt{3(47) + 3} - 4 = 8\]Calculate inside the square root:\[3(47) + 3 = 141 + 3 = 144\]Take the square root:\[\sqrt{144} = 12\]Check the equation:\[12 - 4 = 8\]The original equation holds true, confirming that \( x = 47 \) is the correct solution.

Key Concepts

Isolating the Square Root

When you are solving an equation involving a square root, such as \( \sqrt{3x+3} - 4 = 8 \), your first goal should be to isolate the square root. Isolating a square root means reorganizing the equation so that the square root term is by itself on one side. This is crucial because it simplifies further steps. In this example, you can isolate the square root by adding 4 to both sides of the equation:

• Original equation: \( \sqrt{3x+3} - 4 = 8 \)
• Add 4 to both sides: \( \sqrt{3x+3} = 12 \)

Doing this keeps the equation balanced, meaning both sides remain equal, which is essential in solving equations. Once the square root is isolated, it becomes easier to handle, making the next steps much simpler.

Squaring Both Sides

After isolating the square root in an equation, the next step is to eliminate it by squaring both sides. Squaring is like taking an equation back to a non-squared form, letting you work with the terms linearly. When you square the equation \( \sqrt{3x+3} = 12 \), you apply the square to both sides:

• Left side: \( (\sqrt{3x+3})^2 = 3x+3 \)
• Right side: \( 12^2 = 144 \)

Thus, the equation \( 3x+3 = 144 \) is obtained. The squaring step is crucial as it transforms the equation into a format that is simpler to solve, helping you focus on the variables without the complicating square root symbol. Remember to handle both sides of the equation equally to maintain balance.

Verifying the Solution

Verifying your solution is an important step to ensure accuracy in your calculations. Once you have solved \( x = 47 \), how do you check if it’s correct? You substitute \( x \) back into the original equation to see if both sides equate. Let's substitute \( x = 47 \) into the original equation:

• Inside the square root: \( \sqrt{3 \times 47 + 3} = \sqrt{141 + 3} = \sqrt{144} \)
• The square root of 144 is 12.
• Check the equation: \( 12 - 4 = 8 \)

Since both sides of the equation equal out, we confirm that \( x = 47 \) is indeed the correct solution. Verification is like double-checking your work and builds confidence that the answer you arrived at is accurate.