Questions and Answers
What does infinitely many solutions mean?
infinitely many solutions
Solve the system: 4x + 3y = 12 and 2x + 5y = 6.
(3, 0)
If x = 482 adults and y = 273 students, what is the scenario about?
There were 755 people watching a Highland Park basketball game.
Find the numbers if x = 2, y = 5, z = -3 and the conditions are given.
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Solve for x in | 3x + 4 | - 1 = 29.
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Solve for x in | x - 2 | = 2x - 10.
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Solve for x in -3 | 4x - 3 | = -27.
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What is the solution set for the inequality | 2x - 3 | < 9?
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What is the solution for the inequality | 3x | < -10?
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What is the solution set for - | 6x + 3 | < 15?
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Find the solution for -2 | 2x - 4 | - 3 > -19.
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What is the solution for 2 + 3(3 - x) < -(2x + 1)?
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What is the solution for 2/5 + 4/5x > 1/15?
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Solve the system by graphing: y + 3 < 2x and 2y < x + 2.
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What are the vertex coordinates for the parabola y = x^2 - 2x - 15?
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What is the y-intercept of the parabola y = x^2 - 2x - 15?
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What are the roots of the parabola y = x^2 - 2x - 15?
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What direction does the parabola y = x^2 - 2x - 15 open?
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What are the vertex coordinates for the parabola y = x^2 - 2x - 15?
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What are the roots of the parabola y = x^2 - 2x - 15?
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Solve the quadratic equation 4x^2 + 8x + 3 = 0.
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Solve the quadratic equation 8b^2 + 10b = 3.
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Solve for x in the equation 4x^2 + 32 = 0.
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Find the solutions for a^3 = 49a.
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Solve the equation 6x^2 - 7x = 5.
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Solve the equation 18u^2 - 3u = 1.
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Simplify the expression (2 + 5i) - (4 - 6i).
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Simplify i^47.
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Simplify the expression (4i - 2)(4i - 2).
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Solve by completing the square for x^2 - 4x - 21 = 0.
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Solve by completing the square for 4x^2 - 8x = -3.
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Solve by completing the square for x^2 - 2x + 2 = 0.
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Solve by completing the square for 4x^2 - 16x + 7 = 0.
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What is the vertex of the parabola y = -4x^2 + 16x - 11?
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What is the axis of symmetry for the parabola y = -4x^2 + 16x - 11?
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What is the width of the parabola y = -4x^2 + 16x - 11?
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What is the domain of the parabola y = -4x^2 + 16x - 11?
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What is the range of the parabola y = -4x^2 + 16x - 11?
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What is the vertex of the parabola y = -(x - 2)^2 + 5?
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What is the axis of symmetry for the parabola y = -(x - 2)^2 + 5?
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What is the width of the parabola y = -(x - 2)^2 + 5?
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What is the domain of the parabola y = -(x - 2)^2 + 5?
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What is the range of the parabola y = -(x - 2)^2 + 5?
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What is the vertex of the parabola y = 3x^2 - 9?
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What is the axis of symmetry for the parabola y = 3x^2 - 9?
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What is the width of the parabola y = 3x^2 - 9?
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What is the domain of the parabola y = 3x^2 - 9?
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What is the range of the parabola y = 3x^2 - 9?
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Write the equation in standard form for the parabola that passes through the points (-2, 10), (-1, 3), (0, 10).
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Identify the vertex of the parabola with vertex at (6, -22).
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What are the roots of the parabola with roots at (11.416, 0) and (.584, 0)?
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What is the domain for the parabola with inequality domain stated as all real numbers?
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What is the range for the parabola with an inequality stated as y > -22?
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What is the interval domain for the parabola?
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What is the interval range for the parabola?
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Study Notes
Infinitely Many Solutions
- Refers to situations in systems of equations where multiple combinations of variable values satisfy all equations.
Solving Linear Equations
- Solution for 4x + 3y = 12 and 2x + 5y = 6 is (3,0).
- A Highland Park basketball game involved 482 adults and 273 students, with total attendance of 755 and ticket sales totaling $1614.50.
System of Equations with Variables
- Three numbers that sum to 4: x = 2, y = 5, z = -3. The second number exceeds twice the first by 1, and the sum of the first and third equals -1.
Absolute Value Equations
- Solving 5 | 3x+4 | -1 = 29 results in x = 2/3, -10/3.
- For | x-2 | = 2x-10, the solution is x = 8.
- For -3 | 4x-3 | = -27, solutions are x = -3/2, 3.
Inequalities and Graphing
- | 2x-3 | < 9 yields -3 < x < 6; graph includes this interval.
- | 3x | < -10 has no solution since absolute values cannot be negative.
- For the inequality - | 6x+3 | < 15, the solution set is -3 < x < 2.
Further Inequalities
- For -2 | 2x-4 | -3 > -19, the solution yields -2 < x < 6.
- A solution set of 2 + 3(3-x) < - (2x+1) results in x being greater than 12.
- Solve the inequality 2/5 + 4/5x > 1/15, yielding x > -5/12.
Graphing and Parabolas
- Solve the system by graphing: y + 3 < 2x and 2y < x + 2.
- Given the quadratic equation y = x^2 - 2x - 15, the vertex is (-1/2, -25/4), y-intercept is (0, -6), axis of symmetry is x = -1/2, roots are (-3,0) and (2,0), and it opens upwards.
Quadratic Formula Solutions
- Quadratic equation 4x^2 + 8x + 3 = 0 solved yields x = -1/2 and -3/2.
- For 8b^2 + 10b = 3, solutions are x = 1/4 and -3/2.
- Solving 4x^2 + 32 = 0 results in complex roots: x = ±2i√2.
Polynomial and Complex Solutions
- For a^3 = 49a, solutions are a = 0, -7, and 7.
- 6x^2 - 7x = 5 yields solutions x = 5/3 and -1/2.
- 18u^2 - 3u = 1 leads to u = -1/6 and 1/3.
Simplifying Complex Numbers
- Simplification of (2 + 5i) - (4 - 6i) gives -2 + 11i.
- i^47 simplifies to -i.
- (4i - 2)(4i - 2) simplifies to -12 - 16i.
Completing the Square
- Completing the square for x^2 - 4x - 21 = 0 yields solutions x = -3 and 7.
- The equation 4x^2 - 8x = -3 results in solutions x = 3/2 and 1/2.
- Completing the square for x^2 - 2x + 2 = 0 provides solutions x = 1+i and 1-i.
- For 4x^2 - 16x + 7 = 0, solutions are x = 7/2 and 1/2.
Vertex Form and Characteristics of Parabolas
- For y = -4x^2 + 16x -11, find vertex (2, 5), axis of symmetry x = 2, opens down, domain (-∞, ∞), and range (-∞, 5].
- The second parabola y = -(x-2)^2 + 5 shares vertex (2, 5) and axis of symmetry x = 2, with the same characteristics.
- For y = 3x^2 - 9, the vertex is (0, -9), opens up, domain (-∞, ∞), range [-9, ∞).
Standard Form for Parabolas
- Transform the equation y = 7x^2 + 14x + 10 through given points to standard form.
- Inequalities related to the parabola roots and domain/range: vertex at (6, -22) leads to inequality domain of all real numbers and range y > -22.
Studying That Suits You
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Description
This quiz features a series of flashcards designed to help you practice key concepts in Algebra II. You'll encounter problems related to solving equations and understanding the concept of infinitely many solutions. Perfect for exam preparation!
