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Chapter 7: Problem 9

Solve. $$ \sqrt{5 x+1}=4 $$

Short Answer

Expert verified
The solution to the equation is x = 3.

Step by step solution

01

Isolate the square root

The equation is already isolated with the square root term on one side: \( \sqrt{5x + 1} = 4 \)
02

Square both sides

To eliminate the square root, square both sides of the equation: \[ (\sqrt{5x + 1})^2 = 4^2 \]
03

Simplify the equation

Simplify both sides after squaring: \[ 5x + 1 = 16 \]
04

Solve for x

Isolate the variable x by subtracting 1 from both sides and then dividing by 5: \[ 5x = 15 \ \frac{5x}{5} = \frac{15}{5} \ x = 3 \]
05

Verify the solution

Substitute x = 3 back into the original equation to check the solution: \[ \sqrt{5(3) + 1} = \sqrt{16} = 4 \ This confirms that x = 3 is the correct solution. \]

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Key Concepts

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

Isolating the Variable
In the equation \( \sqrt{5x + 1} = 4 \), the goal is to solve for the variable \((x)\). The first step in solving square root equations is isolating the square root term. In this case, the square root is already on one side of the equation. Here, the term \( \sqrt{5x + 1} \) is isolated, which means there are no other terms on the same side. This is critical because only then can we effectively eliminate the square root by squaring both sides.
Squaring Both Sides
To remove the square root and simplify the equation, square both sides. When we square \sqrt{5x + 1}\ we get: \[ (\sqrt{5x + 1})^2 = 4^2 \] which simplifies to: \[ 5x + 1 = 16 \] Squaring both sides gets rid of the square root sign and makes the equation easier to solve. Be cautious while squaring as it can sometimes introduce extraneous solutions, which we'll address later in the verification step.
Simplifying Equations
After squaring both sides, the next step is simplifying the resulting equation. From \5x + 1 = 16\, we need to isolate \x\ by performing basic algebraic operations. Begin by subtracting 1 from both sides: \[ 5x = 15 \] Then, divide both sides by 5 to solve for x: \[ x = \frac{15}{5} = 3 \] These steps help to isolate the variable, resulting in the solution \x = 3\.
Verifying Solutions
Verification is vital to ensure the proposed solution is correct. Substitute \x = 3\ back into the original equation to verify: \[ \sqrt{5(3) + 1} = \sqrt{16} = 4 \] This shows that the left-hand side equals the right-hand side, confirming that \x = 3\ is indeed correct. Ensure to check your answer as squaring both sides can sometimes introduce false solutions that don't satisfy the original equation.

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