Problem 13 - Elementary and Intermediate Algebra

Question

Complete each solution. Solve: \(\begin{aligned} \sqrt{3 x+3}-1 &=5 \\ \sqrt{3 x+3} &=\\\\(\sqrt{3 x+3}) &=(6) \\ &=36 \\ 3 x &=\\\ x &=\end{aligned}\)

Step-by-Step Solution

Answer

The solution is \( x = 11 \).

Step 1: Isolate the Square Root

Start with the equation \( \sqrt{3x + 3} - 1 = 5 \). Add 1 to both sides to isolate the square root, resulting in \( \sqrt{3x + 3} = 6 \).

Step 2: Eliminate the Square Root

Square both sides of the equation to eliminate the square root. This gives you \( (\sqrt{3x + 3})^2 = 6^2 \), which simplifies to \( 3x + 3 = 36 \).

Step 3: Solve for x

Subtract 3 from both sides to isolate \( 3x \), resulting in \( 3x = 33 \). Then, divide both sides by 3 to solve for \( x \), giving \( x = 11 \).

Key Concepts

Isolating Square Roots

When solving an equation involving a square root, the first step is to isolate the square root on one side of the equation.
This allows you to clearly identify the term under the square root, making it easier to manipulate the equation in future steps.

• Start by moving other terms away from the square root by performing the inverse operation. In our exercise, the square root is involved in the equation \( \sqrt{3x + 3} - 1 = 5 \).
• To isolate the square root, add 1 to both sides: \( \sqrt{3x + 3} = 6 \).

Once the square root is isolated, you are prepared to eliminate the square root in the next step. This process is essential as it simplifies the equation and helps move towards the final solution without complications.

Eliminating Square Roots

After isolating the square root, the next task is to eliminate it. This makes the equation solvable using standard algebraic operations.
The way to achieve this is by squaring both sides of the equation.

• In our scenario, the isolated equation is \( \sqrt{3x + 3} = 6 \).
• Square both sides to remove the square root: \( (\sqrt{3x + 3})^2 = 6^2 \) yields \( 3x + 3 = 36 \).

By squaring, you remove the square root and transform the expression into a basic linear equation.
This step is crucial because it simplifies the equation so you can use basic algebra to solve for the unknown variable.

Algebra Problem Solving

Once the square root is eliminated, you are left with a linear algebra problem that can be solved using simple operations.
The objective is to solve for the variable, in this case, \( x \).

• Start by isolating the term with the variable. With \( 3x + 3 = 36 \), subtract 3 from both sides to obtain \( 3x = 33 \).
• Next, divide each side by 3 to solve for \( x \): \( x = 11 \).

By breaking down the problem into smaller steps, you eliminate complexities and errors. Using such problem-solving strategies in algebra not only helps in arriving at the correct answer but also enhances your understanding of mathematical processes.
Practicing these steps in different problems helps you become more proficient and confident in tackling algebraic equations.