Practice substitution with an odd integer to test answer choices, identify even results, and use elimination to conclude that choices D and F are correct.

Use substitution with integer values for n to test each choice for evenness, rule out options that yield even numbers, and confirm D as the non-even answer.

Use substitution to test fractions greater than 1 and eliminate options in GRE math problems, concluding with choice D.

Test an odd integer n by substitution to determine which choice yields an even result; elimination of A, B, D, and E leaves C as the correct answer.

Substitute x = 4 and y = 9 to test each choice for a perfect square, concluding that D is not a perfect square.

Nova's GRE math prep uses substitution: choose k between 0 and 1 as a perfect square (k=1/4), compare 3/8 to 1/4 by cross-multiplication, and eliminate to show e.

Plug in each answer choice to check how many pages are read, eliminate mismatches, and confirm the correct option, which is option e.

Practice substitution with an even m in m+1, identify the next two even integers, and use value checks (m=2, m=0) to eliminate options and confirm option c.

If x squared is even, x must be even; taking square roots gives x = ±2, which eliminates choices A and C, confirming B as the answer.

Apply substitution with p=1 and q=2 to count integers between 2 and 4, analyzing options to determine the correct choice by elimination.

Demonstrate solving a substitution problem with primes by testing pairs (X, Y) to compute X minus Y, eliminating answer choices, and identifying option C as correct.

Master substitution in quantitative comparisons by testing negative numbers, fractions, zero, and one, using the ordered values 0, 1, 2, -2, and one half.

Learn substitution strategies for quantitative comparisons, recognizing when substituting different numbers yields conflicting results, signaling a double case and that the answer is D.

Master the floor function and substitution for quantitative comparisons by comparing sqrt(x) with floor(x) and identifying cases where information is insufficient.

Master substitution in quantitative comparisons and recognize that not enough information (choice d) can be as likely as a, b, or c on GRE problems.

Demonstrates substitution in quantitative comparisons: for m>0, m=1 yields A=B=1, but m=2 gives 2^100 much larger than 2^10, so not enough information to decide.

Testing negative x values shows column a exceeds column b; squaring yields positives, odd exponents preserve negativity, and negative times negative is positive, so a is correct.

Explains substitution (quantitative comparisons) by choosing x between -1 and 0, using invert-and-multiply to compare negative fractions, and showing negative one half is greater than negative two.

Use substitution to compare columns a and b: with 0, both columns are equal; then with 2 and 1, column b is greater, a double case, and answer is d.

Apply substitution in a quantitative comparison by setting x and y equal to simplify. With x=y=1, column a equals column b; with x=y=2, they differ, indicating not enough information (D).

Use substitution with a negative number to compare columns; equality at a = -1 and inequality at a = -2 reveal not enough information to decide (D).

Explore substitution in quantitative comparisons: if x = y ≠ 0, then x/y is positive, making column b greater than column a.

Apply substitution to a comparison problem by testing x=0 and x=1, evaluating cubes and nearest multiples of ten in columns A and B, and determine that the answer is D.

Explore substitution in quantitative comparisons through problem 10, analyzing column A and B with radical expressions and x-value scenarios to identify the correct option.

Plug-in, or substituting the actual answer choices into the problem, offers an effective alternative to guessing numbers, though it is less common than standard substitution.

Plug in answer choices to satisfy the digits-sum rule and eliminate options that don’t sum to 18, then use the remaining criteria to confirm the correct option.

Apply substitution (plugging in) to a two-digit number where the tens digit is twice the units digit, using elimination of choices to confirm the correct option d.

Practice substitution (plugging in) problems by testing answer choices to make the expression equal to one, using elimination to identify the correct option, as shown with option D using two.

Use the sum of digits equal to 12 to eliminate options, then apply that the 10th digit is one third the units digit to confirm the correct choice as D.

Explore substitution (plugging in) strategies by testing choices and using backward reasoning to solve a bus stop halving problem, revealing the correct option and solution steps.

Learn how to solve substitution problems by plugging in answer choices to verify values, eliminate options, and confirm that the expression evaluates to one.

Learn to solve substitution problems by plugging in answer choices to determine which yields negative one, quickly eliminating options to identify the correct choice, demonstrated with option b.

Replace x with two and y with three in the given formula to obtain three, and enter three into the box in the defined functions example.

Show how defining b as a square makes column a and column b both equal to z squared, so the columns match and the answer is C.

Apply the two-part definition by testing parity with k=1, which yields an odd result, then multiply the inside by 4 to obtain 8k-4 (choice C).

Explore how substitution works in defined functions, using innermost parentheses, replacing x with negative pi and then with 2 pi to determine the answer.

Apply a two-part definition to determine the parity of u and v using contradiction, proving u is odd and v is even, then substitute to obtain a difference of five.

Analyze how base-exponent placement affects outcomes, simplify negative exponents by reciprocals, and verify that choice B is true while C is false.

Identify the base and exponent in X^Y and (X^Y)^Z, with X as the base, Y the exponent, and Z the outer exponent when parentheses raise the entire quantity.

Solve for y in the defined functions 2 example 8 by setting x*y equal to -x, simplify to y^2(x^2+1)=0, apply the zero product property, and conclude y=0.

Learn that the area of a square with side x equals x squared, and see how 81 divided by 9 equals 9, confirming that choice B is correct.

Solve defined functions problem 1 by substituting p = 3 into the given formula, evaluating the resulting expression, and finding the final answer E.

Tackle defined functions problem 2 by analyzing GRC multiple-choice ordering, testing values against two, and plugging 6 into the formula to confirm the answer is a.

Apply the area formula pi r^2 with r = D/2, substitute into the expression to obtain 36 pi^2, and conclude that the correct choice is D.

Resolve a function style problem by substituting values, using the difference of squares to factor a squared minus one as (a+1)(a-1), and conclude that choices b and c are true.

Define x star y as x over y and substitute into the expression. By inverting the denominator and multiplying, (x/y)/z simplifies to x/(yz), the correct choice.

Apply the defined functions approach; compute the square root of 64, divide by 2, then take the square root of 4 and divide by 2 to get 1.

Apply the Pythagorean theorem to a right triangle, find height, and compute area with base 3 and height 4; area is 6, showing column A (10) exceeds column B.

Use the triangle angle sum of 180 degrees to solve for C; C equals 30, B equals 30, Komei is at 30, so the columns are equal.

Use cross multiplication to compare fractions, since the larger product signals the larger fraction; for example, 15/16 is greater than 7/9.

evaluate the fraction by inverting the denominator and multiplying, proving that one over one half equals two, then add one to get three and compare columns to select a.

Treat a ratio as a fraction and solve for x by balancing the fractions; then compare five sixteenths to one over twenty via cross multiplication to decide which is larger.

Observe that taking the square root of a proper fraction between 0 and 1 increases its value, while squaring it decreases it, as shown by 7/8.

Explore how negative values behave under the fourth power and how even exponents negate negatives, demonstrating that two column expressions are equal in a GRE math problem.

Convert decimals to a fraction using invert-and-multiply, compare 25/24, and show that column a is larger than column b, so the answer is a.

Analyze a GRE math problem by comparing fractional powers and roots, using pi ≈ 3.14 to order values and identify the larger option.

Analyze how equal and non-equal x and y affect column values in a GRE math problem, showing when the columns match and identifying a double case with answer D.

Divide column a by r to get s equals 4 over r, and divide column b by t to get s equals 10 over t; reduce to s, proving equality.

Analyze statements about a, b, c as consecutive integers, where one is divisible by 3 and 4. Use parity reasoning to show statement 3 is false; the answer is b.

Apply prime number reasoning and the process of elimination to determine suitable values for X and Y in a gre-style number theory example.

From the innermost parentheses out, simplify the expression inside and outside the absolute value, noting negative behavior. Then apply negation to get x equals three, identifying option C.

Analyze negative products in Nova's GRE math prep course, number theory 4 example 6; compare 14 times each option, eliminate a and b, and select c and d.

Analyze how remainders reveal odd and even forms, convert expressions to standard forms such as n=4v+3 and 2x+1, and verify parity to determine the correct choice.

Show that primes X and Y greater than two are odd, so their product XY is odd and not divisible by 2, yielding the correct choice a.

Explain divisibility by 2 and 5, showing that 5x and 5y are multiples of 10, and that the answer is E.

Identify numbers between two and eight hundred that are both cubes and squares; among 8, 27, 64, and 729, only 64 and 729 are perfect squares.

Evaluate a two-digit number by testing answer choices to meet digit sum four and digit difference four, eliminating options and confirming choice c as the solution.

Explore the remainder concept in number theory, derive P = 9Q+1, test options, and confirm only statement 3 is true.

Identify the prime in column a between 1 and 4, reducing n to 2 or 3; factoring 6 with negatives yields x = 2 or 3, answer d.

Apply a quick fractional comparison using lcd 120 to cancel 1/5 and 1/8, yielding column a as one third and column b as one fourth, showing a larger column a.

Cancel y^4 from both columns to reveal y in column a and -2y^2 in column b; since y>0, divide by y^2 to compare, confirming column a is larger.

Subtract one from both sides of the inequality to show x and x minus one are positive, then multiply the columns to clear fractions and conclude column b is larger.

Divide by x^2 and multiply by x to compare columns; A becomes 1 and B becomes x, so A > B for 0 < x < 1.

Square both columns to eliminate radicals, apply foil, and simplify to 3 plus 5 plus 2√15; add like terms to obtain a plus 2√15, showing the sum exceeds 8.

Multiply both columns by 15 to clear fractions, then square to eliminate radicals. Column A is larger, since 50 is greater than 36.

Set x and y positive and equal to 1 to show A and B are equal; then set x and y equal to 2, making B larger, illustrating double case.

Explore quantitative comparisons by evaluating when x differs from y and when x equals y. Show columns a and b equal 4 in the equal case, illustrating a double case.

Solve GRE quantitative comparison by substitution for negative K; compare column A and B via products of two positive squares, noting a double case at K = -3/2, answer D.

Explore quantitative comparisons by testing x=2,3,4 to compare columns A and B, using smallest positive factors, and show that each case yields equality, leading to answer C.

determine the value of column a and compare it with column b using the order of operations; conclude that column a is larger.

Explore quantitative comparisons by evaluating fractions like one ninth of ten and nine tenths, comparing them to one, and determining the correct option a.

Practice quantitative comparisons by analyzing column a and column b: an odd number of negatives yields a negative, while zero times anything is zero, so column b is larger.

Compare negative and positive fractions minus one to one, excluding zero; squaring minus one half and positive one half yields one quarter, so column b is larger than column a.

Use substitution to compare column a and column b in a GRE quantitative comparisons problem, showing that x=y=1 makes the averages equal, while x=y=2 makes column b larger, answer d.

View a quantitative comparison as an inequality and manipulate both columns by adding, subtracting, or multiplying by a nonzero number. Since y > 0, column A is larger.

Square both columns A and B to remove the radical since the terms are positive. Conclude that column B is larger based on tens and hundreds digits.

Analyze a quantitative comparison by testing the first three positive integers, showing n=1 yields -1, n=2 yields 1, creating a double case, so the answer is D.

Divide both columns by 10 in quantitative comparisons to avoid lengthy multiplication, showing both columns equal as 35 times 54, so the correct choice is D.

Explain quantitative comparisons where A and B are negative; the product and quotient of two negatives are positive; therefore column A is larger.

Cancel y^3 from both columns to get y^2 versus -y. Since y is positive, dividing by y preserves the inequality, so column A is larger.

Clear fractions by using the lcd of 90 to compare column a and column b. Column a equals 80 and column b equals 81, so the answer is b.

Subtract cubes from both columns for a comparison, using negative five for column a and negative fifteen for column b; conclude negative five is greater, so the answer is a.

Analyze x as a positive integer and compare prime counts in column A and B for 1, 2, and 3, showing equal counts and confirming answer C.

Multiply both columns by sqrt(5) to clear the fraction in column A; both columns equal 10, so the correct answer is C.

Analyze a GRE quantitative comparison by examining when X >= 11 makes the columns equal, and when X > 11 makes column A greater than column B, yielding option D.

Master quantitative comparisons by clearing fractions with x minus y and showing x squared minus two xy plus y squared equals (x minus y) squared, proving column equality.

Divide both columns by a (a>0) to clear fractions, compare the resulting values, and determine the first column is larger, giving the answer a.

Apply substitution to compare columns A and B, showing equality when X = Y = 1 and divergence when X ≠ Y, concluding with the double-case result D.

Evaluate the quantitative comparison between column A and column B. With P ≤ 8, the columns are equal when P = 8, and column B is larger when P < 8, giving answer D.

Identify placements in a quantitative comparison, evaluate inside the absolute value, and apply that a negative times a negative is positive. The outer negative makes -5, so columns are equal.

Apply a right triangle concept to GRE quantitative comparisons, using the hypotenuse as the longest side and the x coordinate of point P to compare column B with column A.

Assess a geometric drawing with unknown triangle dimensions and angles to determine that x is greater than y+z; by considering an equilateral triangle, y+z becomes twice x, confirming option d.

Use substitution to compare: set A to 1, A becomes (-1)^2 = 1 > 0; test A = -1, which yields -1 < 0, indicating a double case; answer D.

Apply squaring to both columns to eliminate the radical and compare results; column A yields about two and a quarter, which exceeds two, so the correct choice is B.

Any product of sixes ends with 6 and any product of fives ends with 5; with the same number of factors, the column a value exceeds the column b value.

This quantitative comparisons problem shows how negative numbers can make column A's average larger than column B's, using a double case and yielding answer D.

Apply diameter properties to compare circle chords, showing that BP is greater than PC and that AP is the other chord; conclude column A is bigger than column B.

Evaluate the first three integers greater than one and compare x + x^3 to x^4, demonstrating that column B is larger in all cases, so the answer is B.

Compare the areas of a rectangle and a tilted parallelogram. Tilting reduces area, so the parallelogram’s area is less than 15, making column a smaller than column b (answer b).

Analyze the quantitative comparison by evaluating overlapping ranges of P and Q on a number line, recognizing a double-case scenario, and arriving at the answer D.

Analyze hard quantitative comparisons by testing x^100 versus x^10, show not enough information, and verify with x=1 and x=2 to illustrate the double-case conclusion.

Examine quantitative comparisons of distinct prime factors between x and 4x. Show that 4x contains all x factors, and that the information may be insufficient to decide which has more.

Learn to solve hard quantitative comparisons by substitution to compare 2x and 1/(2x), expose a double case, and conclude the answer is D.

Analyze hard quantitative comparisons by comparing column a and column b with exponents 4 and 3; show why column b is larger in the x=2 case, yielding d.

Expand both expressions to compare columns A and B; A equals sixty three (three squared times seven) and B equals twenty seven (three cubed), so B is larger.

Explore why, for an even X, multiplying by 4 does not introduce new prime factors, so X and 4x have the same number of prime factors, giving answer C.

Use substitution to compare columns in a hard quantitative comparisons problem, testing q>1 with q=2 and other cases; identify smallest factors and show the columns are equal, giving answer c.

Explore geometric reasoning for hard quantitative comparisons by drawing lines to create multiple regions. The example shows how five regions in the shaded area establish that region A is larger.

Examine the six possible intersections between a triangle and a circle, and compare them to three to determine which count is larger.

Evaluate a hard quantitative comparison by analyzing how class composition shifts the weighted average toward 72 or 70, considering cases with more girls or more boys, concluding with answer D.

Explain how a triangle's increasing base can outpace a rectangle's perimeter of similar size in hard quantitative comparisons, noting area equivalence and perimeter as the deciding measure.

Analyze why parallel and congruent line segments ab and cd may appear equal when aligned, but offset drawing makes CM longer than DRM, creating a case; the answer is D.

Explore the floor function to compare values in a hard quantitative comparison, showing that floor(3.1)=3 and floor(-3.1)=-4, so -4 is less than 0.

Apply the Pythagorean theorem to the original triangle (legs 8 and 10) to obtain x = 6, then h = sqrt(51) and y = 8 − sqrt(51) (less than 1).

Translate the straight-angle relation A and B sum to 180 into an equation, set A = 7B, and solve 9B = 360 to find B = 40.

Apply vertical angles to set equal angles and use the straight-line sum of 180 degrees to solve for y; compare 84 and 90 to identify the larger angle.

solve for x using the right angle triangle, then determine z and y as vertical angles, apply angle sums to find w, confirming the solution as a.

Sum the side lengths to find the perimeter, as shown in the example; adding 4, 5, 4, 4, and 3 yields 20.

Set cube volume equal to its surface area, solve e^3 = 6e^2, factor to e^2(e-6)=0, exclude e=0, and obtain edge length six.

Subtract the circle area from the rectangle to get the shaded region; with a rectangle area of 15, and circle area pi r^2 (r=1), the answer is B.

Calculate the shaded region area using outer radius three and inner radius one, giving eight pi. Compare the shaded-to-small-circle area ratio, eight to one, and identify option c.

Explore geometry by equating a square's diagonal with a circle's radius, using a radius of 2 to determine the correct choice D.

Apply the Pythagorean theorem to a right triangle to solve for y, yielding y^2 = 27 and y = sqrt(27), which is greater than five, so column A is larger.

Calculate circle P’s area as pi times 1 squared and circle Q’s area as pi times (1/2) squared. Subtract to get 3pi/4 as the shaded region.

Four arcs centered at vertices form a full circle of radius 3. Subtracting 9 pi from 36 gives the shaded region, so the answer is C.

Explore a geometry problem linking circle area to its radius via r^2 and square roots, determine the square side, and conclude with the final answer.

Apply the triangle angle sum of 180 degrees to relate s, t, and y. With t = 51 degrees, y = 78 degrees, confirming option D.

Apply the triangle angle-sum property to set up an equation, combine like terms, and solve for x; x equals 25, so the correct answer is a.

Solve a geometry problem using parallel lines and corresponding angles, apply the 360-degree sum around a point to set 5x + x = 360, and find x = 60.

Shows x equals 60 degrees in the triangle, solves s = 180/7, finds s < 60, and concludes column B is larger, so the answer is B.

Apply the Pythagorean theorem to the right triangles in the larger and smaller triangles. Solve PS = 6 and QS = sqrt(61) to identify the answer B.

Derive y = 50 − x from angle poq = 70, use x > 50 to obtain 50 − x < 35, inequality flip, and conclude column b is larger.

Calculate the shaded region by subtracting the smaller triangle's area from the larger triangle's area; larger area is 2 with base equals height, smaller area is 9/8, yielding 7/8.

Choose radii 1 and 2 for the circles, compute areas pi and 4pi, find the shaded region 3pi, and state the ratio 3:1 to pick the correct option.

Determine triangle areas from base and height, finding PQS area 15, subtract from 40 to get 25 for PQR, concluding with option D.

In a square, decompose into two right triangles with base 2 and height 4, each area 4, sum to 8 unshaded; subtract from 16 to obtain final area 8.

Use the circle's area 9π to deduce radius 3, then compute circumference 6π and arc length π/2 for a 30° sector; the perimeter is 6 + π/2 (option B).

Explore geometry cases for a chord AC, show AC may not be a diameter, and that x can exceed, equal, or fall below 45 degrees, leaving not enough information.

This lecture examines symmetric drawings where angles p, q, and r are equal in a case, and where q plus r equals 180, yielding equal columns and a double case.

Explore geometry extreme cases of side BC, considering BC less than seven and BC greater than seven, revealing a double-case scenario with answer D.

Analyze an isosceles triangle with unknown side AC, demonstrating extreme-case reasoning and why not enough information to decide leads to ambiguity in GRE geometry problems.

Analyze two geometry drawings where theta is less than 45 degrees in case one and greater than 45 degrees in case two; this double-case yields the answer D.

Analyze triangle area relationships using base and height, compare cases where areas are 8 and the square root of 15 against 4.5, and identify the double-case solution D.