Problem 46 - Algebra 2

Question

ACT/SAT If \(\sqrt{x+5}+1=4,\) what is the value of \(x ?\) $$ \begin{array}{lllll}{\text { A } 4} & {\text { B } 10} & {\text { C } 11} & {} & {\text { D } 20}\end{array} $$

Step-by-Step Solution

Answer

The value of \(x\) is 4.

Step 1: Isolate the Square Root

To solve the equation \(\sqrt{x+5} + 1 = 4\), we first need to isolate the square root on one side. We can do this by subtracting 1 from both sides: \[ \sqrt{x+5} = 4 - 1 \] which simplifies to \[ \sqrt{x+5} = 3. \]

Step 2: Eliminate the Square Root

To get rid of the square root, square both sides of the equation. This gives us: \[ (\sqrt{x+5})^2 = 3^2 \] \[ x+5 = 9. \]

Step 3: Solve for x

Now, solve for \(x\) by subtracting 5 from both sides of the equation: \[ x + 5 - 5 = 9 - 5 \] \[ x = 4. \]

Step 4: Verify Solution

Substitute \(x = 4\) back into the original equation to verify the solution. Check if \(\sqrt{4+5} + 1 = 4\). This simplifies to \(\sqrt{9} + 1 = 4\), which simplifies further to \(3 + 1 = 4\). Since both sides equal, \(x = 4\) is the correct solution.

Key Concepts

Solving Square Root Equations

When tackling square root equations, the goal is to remove the square root symbol and solve for the variable inside. For example, if you start with an equation like \( \sqrt{x+5} + 1 = 4 \), the first step is to isolate the square root itself. This helps simplify the problem and set you up for the solution.
By manipulating the equation, you can transform it into a simpler form without the square root. Simply subtract numbers on both sides: \( \sqrt{x+5} = 3 \).
Next, eliminate the square root by squaring both sides of the equation. That means making each side an exponent of two. The operation transforms \((\sqrt{x+5})^2 = 3^2\) into \(x+5 = 9\).
So, remember these steps:

• Isolate the square root.
• Square both sides to remove the square root.

Once you remove the square root, the path to finding the variable becomes much clearer.

Isolating Variables

Isolating variables is a fundamental part of solving equations, including those involving square roots like in this example. When you isolate a variable, you're working to get that variable by itself on one side of the equation. This simplifies the process of solving the equation.
For instance, with an equation like \( x + 5 = 9\), your objective is to find the value of \( x\). To do this, perform operations that will eliminate all other numbers on the left side. Here, it involves subtracting 5 from both sides to solve for \( x \). So, \( x = 9 - 5 \) simplifies to \( x = 4 \).
When isolating variables:

• Perform the same mathematical operation on both sides of the equation.
• Keep your goal in mind: to get the variable alone on one side.

Always be thorough and double-check your work to ensure your variable is correctly isolated.

Verifying Solutions

After solving an equation, especially one involving roots, it's essential to verify your solution to ensure accuracy. Verification means plugging your value back into the original equation to check its correctness.
In this scenario, once you find \( x = 4 \), you should substitute it back into the initial equation \( \sqrt{x+5} + 1 = 4 \) as follows: Calculate \( \sqrt{4+5} + 1\). This process simplifies to \( \sqrt{9} + 1 = 4 \). Further simplifying, \( 3 + 1 = 4 \), which is true!
So, some tips to remember when verifying solutions:

• Substitute the solution back into the original equation.
• Ensure both sides of the equation are equal after substitution.

Doing this provides peace of mind that your answer is correct and confirms your understanding of the problem.