Questions and Answers
What is the total number of possible passwords calculated from the given numbers?
28,800
If the least common multiple of 9, 10, 12, and v is 540, which of the following could be v? (Choices: A. 24, B. 18, C. 27, D. 45, E. 36)
What is the value of abc for the equation y=axb+c given the coordinate pairs (0, 2), (1, 7), and (2, 42)?
30
Which of the following numbers completes the sequence 3, 8, 14, 21, 29, ___?
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How long did Beth take to paint each room if she could paint 5 identical rooms during one 6-hour shift?
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If 5x + 3 = 10x - 17, what is the value of x?
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What is the sum of the three factors of k if k is the next-highest integer after 4 that has exactly three factors?
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What was William's average monthly pay for the 12 months?
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What number results from multiplying by 4 and then subtracting 7, equating to subtracting 7 then multiplying by 11?
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The answer to the inequality ______ is:
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What is the expression for y in the similar triangles problem?
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Which of the following is equal to (n - 3)² if n = 11?
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What is the diagonal distance across a square field with an area of 22,500 square feet?
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What is the slope of a line passing through points (1, 3) and (4, -3)?
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How many squares of sod does Andrea need to buy for her two sections measuring 30 x 40 feet and 60 x 80 feet?
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What is the value of p given pq - 3r = 2 and q - r = 7?
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If x^2 - x - 2 > 0, what is the solution set for x?
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What is the remaining corner of a parallelogram with three known corners at (3, 3), (4, -4), and (-2, -1)?
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What was Matt's speed if Doug ran faster and passed him after completing six laps?
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What is the value of mn if the equation x^2 + mx + n = 0 has solutions x = k and x = 2k?
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How many different passwords can Eloise create under her constraints?
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Study Notes
Sequences and Patterns
- The sequence 3, 8, 14, 21, 29 increases by successively larger integers: 5, 6, 7, 8, leading to the next number being 38 (29 + 9).
Rates and Time
- Beth painted 5 rooms in 360 minutes, taking 72 minutes (or 1 hour and 12 minutes) to paint each room.
Solving Linear Equations
- For the equation 5x + 3 = 10x - 17, solving gives x = 4.
Factors and Integers
- The next highest integer after 4 with exactly three factors is 9, which has factors 1, 3, and 9. Their sum is 13.
Average Monthly Income
- William's average monthly pay over 12 months is $3,000, calculated from his $32,000 as a teacher and $4,000 as a barista.
Algebraic Manipulation
- The equation derived from "multiply by 4 and subtract 7" equals "subtract 7 then multiply by 11" leads to the solution x = 10.
Inequalities
- The solution set for -5 ≤ x < 5 represents all values between -5 and 5, inclusive of -5 but not 5.
Similar Triangles
- In similar triangles, if the side lengths of the smaller triangle are 3 and 2, the corresponding large triangle sides are 6 and 4, leading to y = x + 4.
Quadratic Equations
- For n = 11, (n - 3)² simplifies to (n + 5)(n - 7) through substitution.
Geometry
- The diagonal distance across a square field with an area of 22,500 square feet is approximately 212 feet, calculated from the side length of 150 feet.
Slope Calculation
- The slope of a line through points (1, 3) and (4, -3) is -2, derived from the change in y over the change in x.
Area and Volume
- Total sod needed for two sections of Andrea's backyard is 1,500 squares, calculated from the total area of 6,000 square feet divided by 4 square feet per square of sod.
Trigonometry
- The secant of an angle relates to the hypotenuse over the adjacent side in right triangles.
Algebraic Isolation
- Isolating terms in pq - 3r = 2 yields a relationship for solving in terms of p and r.
Arithmetic
- The final answer for a simple fraction problem is 1/2.
Sum of Integers
- The product of two integers plus 4 equaling 40 leads to sums, with 18 identified as impossible from combined pairs.
Angle Relationships
- The sum of opposite angles on parallel lines (∠p and ∠r) equates to 180°.
Solutions to Quadratics
- The values a = 4 and b = 2 in t² - 6t + 8 = 0 result in a - b being 2.
Box Dimensions
- A box with a volume of 135 cm³ and sides measuring 3 cm and 9 cm results in the area of the largest side being 45 cm².
Intersection of Lines
- Lines represented by 3x + 2y = -2 and 5x - y = 14 intersect at the point (2, -4).
Polynomial Factoring
- The expression 12x²y³z⁴ - 30xy² + 24x²y⁵z factors to 6xy²(2xyz⁴ - 5 + 4xy³z) based on the greatest common factor.
Phone Plan Costs
- Anderson's phone plan after 1,000 minutes costs modeled by the equation f(m) = 0.25m - 200.
X-Intercept Calculation
- A line with slope 2 passing through (3, 4) has an x-intercept of 1.
Saving Money
- Anne and Katherine's combined savings total $750, solved through successive relationship equations regarding their amounts.
Absolute Value Inequality
- The inequality |2x + 7| > 11 yields solutions of x < -9 or x > 2.
Distance Formula
- The distance between points (-2, 4) and (1, -2) can be calculated using the distance formula.
Math Expectations
- The problem set engages varying skill levels across algebra, geometry, trigonometry, and word problems with algebraic applications.
Cube Relationships
- A cube with volume k cm³ and surface area of 10k cm² has a height of 3/5.
Complex Numbers
- The expression (4 + 2i)(4 - 2i) simplifies to 20 using the FOIL method and properties of i² = -1.
Meal Cost Calculation
- Sebastian's meal cost without a 15% tip amounts to $30.60, calculated from the total payment.
Appointment Timing
- Andrea's arrival time before a registered delay was determined as 11:23.
Solutions to Inequalities
- The inequality x² - x - 2 > 0 has solutions of x < -1 or x > 2 based on the factorization and parabola behavior.
Parallelogram Coordinates
- The coordinates for the missing corner of a parallelogram are solved and determined to be (-3, 6).
Track Speed Problem
- Matt's speed in a track race is determined to be 10 miles per hour based on comparison speeds with Doug.
Quadratic Relationships
- For the equation structure yielding solutions of x = k and x = 2k, mn resolves to -6k³ through systematic equation building.
Password Combinations
- Eloise's password formulation follows a strict set of criteria resulting in approximately 28,800 possible valid passwords.
Least Common Multiple
- To find v such that the LCM of 9, 10, 12, and v equals 540, the only possible solution identified is 27.
Polynomial Values
- The equation y = ax^b + c leads to the resultant abc value being 30 through various substitutions and calculations.
Studying That Suits You
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Description
Test your knowledge with these algebra flashcards focused on sequences and number patterns. This quiz will help reinforce your understanding of arithmetic sequences and the concept of adding sequentially larger numbers. Perfect for preparing for your Algebra II final exam.
