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 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 for the expression equivalent to $√(9a^2$) + $√(16a^2$) for a ≥ 0. Simplify each: $√(9a^2$) = 3a, $√(16a^2$) = 4a, so 3a + 4a = 7a. This assumes a ≥ 0 to avoid absolute values. Options like 25a might come from multiplying instead of adding. A key error is treating them as $√(9a^2$ + $16a^2$), which is incorrect. Verify: for a=1, √9 + √16 = 3+4=7, matches 7a. When adding simplified radicals, handle each separately before combining.

The question requires solving |2x+3| = |x-1| considering all sign cases. Square both sides: $(2x+3)^2$ = $(x-1)^2$, $4x^2$ +12x+9 = $x^2$ -2x+1, $3x^2$ +14x +8=0. Solutions: x = [-14 ± √(196-96)]/6 = [-14 ±10]/6, so x=-4 or x=-2/3. Check: for x=-4, | -8+3|=5, | -4-1|=5; for x=-2/3, | -4/3+3|=5/3, | -2/3-1|=5/3. Both work. A common error is missing a solution by not squaring properly. Always verify in original after solving.

To solve the absolute value equation $|2x-7| = 9$, we need to consider that the expression inside the absolute value bars can equal either 9 or -9. This gives us two cases: Case 1: $2x - 7 = 9$, which yields $2x = 16$, so $x = 8$. Case 2: $2x - 7 = -9$, which yields $2x = -2$, so $x = -1$. We can verify: when $x = 8$, $|2(8)-7| = |16-7| = |9| = 9$ ✓, and when $x = -1$, $|2(-1)-7| = |-2-7| = |-9| = 9$ ✓. The most common error is solving only one case and forgetting that absolute value represents distance, which can be achieved in two directions. Remember that $|A| = k$ means $A = k$ or $A = -k$ when $k > 0$.

This problem asks us to simplify an expression with three radical terms by combining like radicals. First, we need to simplify each radical by factoring out perfect squares: $\sqrt{72} = \sqrt{36 \cdot 2} = 6\sqrt{2}$, $\sqrt{8} = \sqrt{4 \cdot 2} = 2\sqrt{2}$, and $\sqrt{18} = \sqrt{9 \cdot 2} = 3\sqrt{2}$. Now we can substitute these simplified forms: $6\sqrt{2} - 2(2\sqrt{2}) + 3\sqrt{2} = 6\sqrt{2} - 4\sqrt{2} + 3\sqrt{2}$. Since all terms contain $\sqrt{2}$, we can combine the coefficients: $(6 - 4 + 3)\sqrt{2} = 5\sqrt{2}$. A common error is trying to combine radicals before simplifying them or incorrectly factoring the numbers under the radicals. When working with radicals on the PSAT, always simplify each radical first before attempting to combine terms.

The question requires simplifying $√(75x^3$) in simplest radical form assuming x ≥ 0. Factor $75x^3$ = 25 * 3 * $x^2$ * x = $25x^2$ * (3x). Take square root: $√(25x^2$ * 3x) = $√(25x^2$) * √(3x) = 5x √(3x). This is simplest as 3x has no perfect squares. A common error is incomplete factoring, like missing $x^2$. Check by squaring: (5x $√(3x))^2$ = $25x^2$ * 3x = $75x^3$, correct. Always factor out perfect squares when simplifying radicals with variables.

The question requires solving the absolute value equation |x - 1| + 2 = 7. Isolate the absolute value: |x - 1| = 5. Consider two cases: x - 1 = 5 gives x = 6, and x - 1 = -5 gives x = -4. Both satisfy the original: |6-1| + 2 = 7, |-4-1| + 2 = 7. A common error is forgetting to isolate before splitting cases. Always check solutions in the original equation to confirm. For absolute value equations, isolate first then handle positive and negative scenarios.