To solve $\sqrt{2x-1} + 1 = 5$, we first isolate the radical by subtracting 1 from both sides: $\sqrt{2x-1} = 4$. Now we square both sides to eliminate the square root: $(\sqrt{2x-1})^2 = 4^2$, which gives us $2x - 1 = 16$. Solving for $x$: $2x = 17$, so $x = \frac{17}{2}$. Let's verify by substituting back: $\sqrt{2(\frac{17}{2})-1} + 1 = \sqrt{17-1} + 1 = \sqrt{16} + 1 = 4 + 1 = 5$ ✓. A common error is trying to distribute operations across the radical before isolating it, such as incorrectly thinking $\sqrt{2x-1} + 1 = \sqrt{2x} + \sqrt{0}$. Always isolate the radical term before squaring both sides.

We need to simplify $\sqrt{12x^2y^3}$ where $x \geq 0$ and $y \geq 0$. First, factor the expression under the radical to identify perfect squares: $12x^2y^3 = 4 \cdot 3 \cdot x^2 \cdot y^2 \cdot y = 4x^2y^2 \cdot 3y$. Now we can take the square root of the perfect square factors: $\sqrt{4x^2y^2 \cdot 3y} = \sqrt{4x^2y^2} \cdot \sqrt{3y} = 2xy\sqrt{3y}$. Note that $\sqrt{x^2} = x$ (not $|x|$) because we're given that $x \geq 0$. A common error is leaving perfect square factors under the radical or incorrectly handling the variable exponents. Always factor completely and extract all perfect squares when simplifying radicals.

The question requires simplifying √98 + √8 in simplest radical form. Factor: √98 = √(492) = 7√2, √8 = √(42) = 2√2. Add: 7√2 + 2√2 = 9√2. This is simplest as like terms are combined. A key error is adding under one radical, like √(98+8), which is wrong. Check by approximating: 7√2 ≈9.899, 2√2≈2.828, sum≈12.727; 9√2≈12.727, matches. Always simplify each radical before combining.

The question requires finding the equivalent expression to 3/(2 - √5) after rationalizing the denominator. Multiply numerator and denominator by the conjugate 2 + √5: 3(2 + √5) / ((2 - √5)(2 + √5)) = 3(2 + √5) / (4 - 5) = 3(2 + √5) / (-1). This simplifies to -3(2 + √5), matching the form in choice B. Verify numerically: original ≈ 3 / (-0.236) ≈ -12.71, and -3(4.236) ≈ -12.71. A key error is using the wrong conjugate or mishandling the negative denominator. Another mistake could be not distributing the negative sign correctly. When rationalizing, always check your work by verifying numerical equivalence.

The question asks for the solution set of the absolute value equation |x - 4| = 9, requiring both solutions. To solve, consider the two cases: x - 4 = 9 or x - 4 = -9, leading to x = 13 or x = -5. The solution set is {-5, 13}. Verify by substitution: | -5 - 4 | = | -9 | = 9, and |13 - 4| = 9, both correct. A common error is only considering the positive case and missing the negative solution. Another mistake might be incorrect arithmetic when solving the linear equations. When solving absolute value equations, always handle both cases and verify solutions to ensure accuracy.

The question asks to solve |x| = |x - 6| interpreting as distances on the number line. This means distance from x to 0 equals distance to 6, so x is midpoint: x = 3. Check: |3| = 3, |3-6| = 3, equal. Algebraically, consider cases: if x ≥ 6, x = x-6 impossible; if 0 ≤ x < 6, x = 6 - x, 2x=6, x=3; if x < 0, -x = 6 - x impossible. Only x=3. A key error is assuming symmetric solutions like ±3. Always check all intervals defined by critical points.

The question requires solving the inequality (|x - 2| \geq 5) and giving the solution as a union of intervals. Consider the two cases for absolute value: x - 2 ≥ 5 or x - 2 ≤ -5, so x ≥ 7 or x ≤ -3. The solution is (-∞, -3] ∪ [7, ∞). Verify boundary points: at x=-3, | -3-2 | =5 ≥5; at x=7, |7-2|=5 ≥5. A key error is reversing the inequalities or forgetting the union, leading to intervals like [-3,7]. Another mistake might be using strict inequalities instead of inclusive. For absolute value inequalities, always handle the two cases separately and test points in each interval to confirm.

The question requires the equation and solutions for a point 5 units from -2 on the number line. The absolute value represents distance, so |x - (-2)| = 5 simplifies to |x + 2| = 5. Solve the two cases: x + 2 = 5 gives x = 3, and x + 2 = -5 gives x = -7. Both satisfy the distance condition. A common error is switching the sign in the equation, like using |x - 2| which would be distance from 2 instead. Verify by calculating distances: from 3 to -2 is 5, from -7 to -2 is 5. When solving absolute values, always consider both positive and negative cases and check solutions.

The question asks for the domain of f(x) = √(7 - 2x) as an inequality. The expression inside the square root must be non-negative: 7 - 2x ≥ 0. Solve: -2x ≥ -7, then divide by -2 (reverse inequality): x ≤ 7/2. This ensures the radical is defined for real numbers. A common error is forgetting to reverse the inequality sign when dividing by negative. Verify boundary: at x = 7/2, √0 = 0, defined; for x > 7/2, negative inside, undefined. When finding domains of radicals, solve the inequality carefully and test points.

The question asks which expression is equivalent to (\d$\frac{\sqrt{5}$ + $\sqrt{20}$}{$\sqrt{5}$}) in simplest form. Split the fraction: (= \d$\frac{\sqrt{5}$}{$\sqrt{5}$} + \d$\frac{\sqrt{20}$}{$\sqrt{5}$} = 1 + $\sqrt{\dfrac{20}{5}$} = 1 + $\sqrt{4}$ = 1 + 2 = 3). Alternatively, simplify ($\sqrt{20}$ = 2$\sqrt{5}$), so (\d$\frac{\sqrt{5}$ + 2$\sqrt{5}$}{$\sqrt{5}$} = \d$\frac{3\sqrt{5}$}{$\sqrt{5}$} = 3). A common error is not simplifying fully, like leaving it as 1 + $\sqrt{4}$, which equals 3 but isn't the simplest form. Another mistake might be thinking it's $\sqrt{25}$ = 5 by incorrect addition. When simplifying radical expressions in fractions, break them down and combine like terms before evaluating.