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Chapter 3: Problem 31

Solve. $$\sqrt{3 x-4}=1$$

Short Answer

Expert verified
x = \frac{5}{3}

Step by step solution

01

Isolate the square root

The given equation is \( \sqrt{3x - 4} = 1 \). This equation is already isolated with the square root on one side and 1 on the other.
02

Square both sides

Square both sides of the equation to eliminate the square root. \[ (\sqrt{3x -4})^2 = 1^2 \] This simplifies to \[ 3x - 4 = 1 \]
03

Solve for x

Add 4 to both sides to isolate the term with x. \[ 3x - 4 + 4 = 1 + 4 \] This simplifies to \[ 3x = 5 \]Next, divide both sides by 3 to solve for x. \[ x = \frac{5}{3} \]

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Key Concepts

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

square root isolation
When solving square root equations, the first step is often to isolate the square root on one side of the equation. In the given problem, \( \sqrt{3x - 4} = 1 \), the square root is already isolated. You want the square root expression by itself, so the next steps can be correctly applied.
squaring both sides
To eliminate the square root in an equation, you need to square both sides. This will give you an equation without square roots, making it easier to solve for the variable. Squaring the isolated square root equation \( \sqrt{3x -4} = 1 \) results in: \[ (\sqrt{3x -4})^2 = 1^2 \] which simplifies to: \[ 3x - 4 = 1 \] \
Make sure to handle both sides of the equation equally to maintain its balance.
solving linear equations
After squaring both sides, you'll usually get a linear equation. In this example, \(3x - 4 = 1\):
  • Add 4 to both sides to move the constant term: \[ 3x - 4 + 4 = 1 + 4 \]. This simplifies to \[ 3x = 5 \]
  • Next, divide both sides by 3 to isolate x: \[ x = \frac{5}{3} \]
Always perform the same operation on both sides to keep the equation balanced. Recheck each step to ensure accuracy.

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