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Chapter 7: Problem 11

Solve. $$ \sqrt{3 x}+1=5 $$

Short Answer

Expert verified
x = \frac{16}{3}

Step by step solution

01

Isolate the square root term

Subtract 1 from both sides of the equation to isolate the term containing the square root on one side. \[ \sqrt{3x} + 1 - 1 = 5 - 1 \ => \sqrt{3x} = 4 \]
02

Square both sides

Square both sides of the equation to eliminate the square root. \[ (\sqrt{3x})^2 = 4^2 \ => 3x = 16 \]
03

Solve for x

Divide both sides by 3 to solve for x. \[ 3x \div 3 = 16 \div 3 \ => x = \frac{16}{3} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Isolating the Square Root
To solve a square root equation like \(\textstyle \sqrt{3x} + 1 = 5\), we first need to isolate the square root term. This means getting the square root expression by itself on one side of the equation. Let's break it down:
\(\textstyle \sqrt{3x} + 1 = 5\)
Here, the \(\textstyle + 1\) is outside the square root. To isolate \(\textstyle \sqrt{3x}\), we subtract 1 from both sides:
\(\textstyle \sqrt{3x} + 1 - 1 = 5 - 1\)
This simplifies to:
\(\textstyle \sqrt{3x} = 4\)
Now the square root term is isolated on one side, and we can move to the next step.
Squaring Both Sides
Once we've isolated the square root, the next step is to get rid of it. We do this by squaring both sides of the equation. Remember, squaring a square root removes the square root symbol:
\(\textstyle \sqrt{3x} = 4\)
When we square both sides, we get:
\(\textstyle (\sqrt{3x})^2 = 4^2\)
This simplifies to:
\(\textstyle 3x = 16\)
Squaring both sides eliminates the square root, leaving us with a simpler equation to solve.
Solving for x
In the final step, we solve for x. We now have the equation:
\(\textstyle 3x = 16\)
To isolate x, we need to divide both sides by 3:
\(\textstyle \frac{3x}{3} = \frac{16}{3}\)
Simplifying, we get:
\(\textstyle x = \frac{16}{3}\)
And that's our solution! To summarize, we first isolated the square root, squared both sides to remove the square root, and then solved for x by simple division. By mastering these steps, you can solve any similar square root equation.

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Most popular questions from this chapter

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