We are given the equation:

$3 + \sqrt{4x - 5} = 8$

Step 1: Isolate the square root term

First, subtract 3 from both sides of the equation:

$\sqrt{4x - 5} = 8 - 3$

$\sqrt{4x - 5} = 5$

Step 2: Eliminate the square root by squaring both sides

Next, square both sides of the equation to eliminate the square root:

$(\sqrt{4x - 5})^2 = 5^2$

$4x - 5 = 25$

Step 3: Solve for $x$

Now, add 5 to both sides to isolate the term with $x$:

$4x = 25 + 5$

$4x = 30$

Finally, divide both sides by 4:

$x = \frac{30}{4} = \frac{15}{2}$

Thus, the solution is:

$x = \frac{15}{2} \quad \text{or} \quad x = 7.5$

Step 4: Check the solution

Substitute $x = 7.5$ back into the original equation to verify:

$3 + \sqrt{4(7.5) - 5} = 8$

$3 + \sqrt{30 - 5} = 8$

$3 + \sqrt{25} = 8$

$3 + 5 = 8$

Since both sides are equal, the solution is correct.

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Would you like further details or have any questions about this process?

Related questions:

1. How do you handle square roots in equations with more complex terms?
2. What are the steps to check if your solution to an equation is correct?
3. Can there be any extraneous solutions when squaring both sides of an equation?
4. How would you solve similar equations with multiple square roots?
5. What happens if the expression inside the square root becomes negative?

Tip: Always check the solution by substituting it back into the original equation to avoid extraneous solutions.