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Chapter 3: Problem 49

Solve. $$\sqrt{2 x+1}-3=3$$

Short Answer

Expert verified
x = 17.5

Step by step solution

01

Isolate the square root term

Add 3 to both sides of the equation to isolate the square root term. So, \( \ \ \sqrt{2x + 1} - 3 + 3 = 3 + 3 \ \ \ \sqrt{2x + 1}=6 \).
02

Square both sides

Square both sides of the equation to eliminate the square root. \( (\sqrt{2x+1})^2=6^2 \ \ \ 2x + 1 = 36 \).
03

Solve for x

Subtract 1 from both sides to start solving for x: \( 2x + 1 - 1 = 36 - 1 \ \ \ 2x = 35 \). Divide both sides by 2: \( x = \frac{35}{2} \ \ \ \ x = \frac{35}{2}=17.5 \).

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Key Concepts

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

Isolating the Square Root
Before solving an equation involving a square root, we first need to isolate the square root term.
This means making sure that the square root is by itself on one side of the equation.
In the exercise, we start with the equation \(\sqrt{2x + 1} - 3 = 3\).
To isolate \(\sqrt{2x + 1}\), simply add 3 to both sides.
Squaring Both Sides
Once the square root term is isolated, the next step is to eliminate the square root.
This is done by squaring both sides of the equation.
Squaring is a straightforward way to remove the square root.
In our example, after isolating, we have \(\sqrt{2x + 1} = 6\).
We square both sides to get \((\sqrt{2x + 1})^2 = 6^2\).
This simplifies to \((2x + 1 = 36)\).
When you square both sides, ensure you apply the operation correctly to avoid errors.
Solving for x
After squaring both sides and eliminating the square root, the next step is to solve the equation for x.
We now have a simple linear equation: \(2x + 1 = 36\).
Start by subtracting 1 from both sides to get \(2x = 35\).
Finally, divide both sides by 2 to isolate x.
This gives us \(x = \frac{35}{2}\), which simplifies to \(x = 17.5\).
Always double-check your solutions by plugging the value of x back into the original equation to confirm it satisfies the equation.

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