Questions and Answers
What are the solutions for the equation $x²-9x+18 = 0$?
What is the average speed for a round trip between two cities if the speeds are 600 miles/hour and 300 miles/hour?
What is $g(f(x))$ if $f(x) = 7x² - 3$ and $g(y) = 2y + 9$?
Comparing Quantity A: x and Quantity B: 2x, which statement is true?
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What is the value of n if one less than three-eighths of n is the third prime integer?
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If $(x - 5)² = 900$, what is one possible value for x?
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What is the value of x if $3x + y = 13$ and $x - 2y = -12$?
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For the equation $(b * b⁴ * b⁷)^{1/2}/(b³ * b^{x}) = b⁵$, if b is not negative, then what is x?
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If $6h - 2g = 4g + 3h$, what is h in terms of g?
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What is the value of x in the equations $4x + 3y = 6$ and $2x + 2y = 4$?
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If Sally is 2 years younger than Abby, Daisy is 5 years older than Tracy, and Abby is 6 years older than Tracy, which quantity is greater?
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Which is greater: Quantity A: $4^3$ or Quantity B: $3^4$?
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If $a = 1/3b$ and $b = 4c$, then what is $a - b + c$ in terms of c?
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What is the value of $2y - 1$ if $x = 4$ and $y = 3x + 5$?
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How can $25x²-36y²$ be factored?
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How much greater is the degree measure of the sector representing workers in the construction industry compared to the financial industry if the values are 18° and 25°?
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If $[(7/8)]^n = \sqrt{([(7/8)]^5)},$ what is the value of n?
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Which is greater, Quantity A: |x| or Quantity B: $x^3$?
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If y is a real number such that $y ≥ 1$, compare the quantities: Quantity A: $(y²)(y⁴)$ and Quantity B: $y⁸$.
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Bill can build k toys in 6 hours and Bob can build k toys in 3 hours. How long would it take them to build 4k toys together?
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If Kim is twice as old as Claire, Nick is 3 years older than Claire, and their combined age equals 81, how old is Nick?
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What is the next consecutive odd integer after the sum of two consecutive odd integers that is equal to 32?
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In a comparison, Quantity A: $x²$ and Quantity B: $x³$, which is greater?
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Given one root of the equation $x² + kx - 12 = 0$ is 3, compare Quantity A: the value of k and Quantity B: -1.
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What digit appears in the units place when $2^{102}$ is multiplied out?
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Which best describes the relationship between $(x+y)³$ and $x³+y³$ if $x, y eq 0$?
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Study Notes
Algebra and Equations
- Quadratic equation: x² - 9x + 18 = 0, roots are x = 3 and x = 6.
- Average speed calculated using the formula for two speeds: (2 * Speed1 * Speed2) / (Speed1 + Speed2) results in an average speed of 400 miles/hour for a jet returning from City 2 to City 1.
- Composition of functions: g(f(x)) results in 14x² + 3 for f(x) = 7x² - 3 and g(y) = 2y + 9.
Quantitative Comparisons
- Comparison of x (Quantity A) and 2x (Quantity B) shows that the relationship cannot be determined as x could be positive or negative.
- Problem involving prime numbers states that if one less than three-eighths of a positive integer n equals the third prime (5), then n is 16.
Solving Equations
- Equation (x - 5)² = 900 indicates potential solutions: x = 35 or x = -25.
- System of equations: Solving 3x + y = 13 and x - 2y = -12 yields value x = 2.
- Expression manipulation reveals if 6h - 2g = 4g + 3h, then h = 2g.
Factorization and Age Problems
- The factorization of 25x² - 36y² results in (5x - 6y)(5x + 6y).
- Age relationships indicate Sally (younger than Abby) and Daisy (older than Tracy) leads to a conclusion that Daisy's age is greater than Sally's.
Exponential and Quantitative Comparisons
- Comparing powers: Quantity A (4³) is less than Quantity B (3⁴).
- Given relationships a = 1/3b and b = 4c, the expression a - b + c simplifies to -5/3c.
Toys Production Problem
- Work rate problem: Bob and Bill together can build 4k toys in 8 hours based on their respective speeds (k toys in 6 hours for Bill and k in 3 hours for Bob).
Additional Challenges
- Finding the next consecutive odd integer after the sum of two consecutive odd integers (sum = 32) leads to identifying the next integer as 19.
- The equation x² + kx - 12 = 0 having one root as 3 determines that k > -1.
Exponents and Digits
- The units digit of 2^102 parallels observing the cycle of units digits in powers of 2, confirming it is 4.
- The relationship between (x + y)³ and x³ + y³ indicates expansion to include terms related to both x and y if neither is equal to 0.
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Test your knowledge of algebra concepts and problem-solving skills with these flashcards. Each card presents a unique algebraic equation or quantitative reasoning problem for you to solve. Challenge yourself and improve your understanding of algebraic principles!
