Questions and Answers
Write an algebraic expression for the phrase 'a number t times 6'.
6t
Solve the equation 4x + 12 = 2x - 6.
x = -9
What is the solution to the equation 3(x - 3) = 3x - 6?
no solution
Write an algebraic expression for '13 more than the product of 9 and n'.
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Solve the equation (b/3) - 1 = 5.
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Solve the equation -2(x + 3) = 4x - 3.
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Simplify the expression 36 ÷ (4 + 2^3).
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Solve the equation 4k - 5k = 10 + k.
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What is the solution to the equation 8x + 6 - 9x = 2 - x - 15?
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Simplify the expression 14 + (9^3 - 5^2).
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Solve the equation 6(2x + 10) = 10(x + 6).
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Solve the equation 6 - 2x - 1 = 4x + 8 - 6x - 3.
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Evaluate the expression 5m + 6n^2 - n^3 when m = 2 and n = 4.
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Solve the equation -2(x - 5) = 10 - 2x.
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Solve the absolute value equation |2x - 4| = 22.
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Solve the absolute value equation |x - 3| - 6 = 2.
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Evaluate the expression (3x)^2 - (x^3 - y^2) for x = -3 and y = -5.
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Write and solve the equation for the perimeter of a rectangle given that length = 3w + 1 and width = w, with a perimeter of 82 cm.
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Simplify the expression (-1/5)^3.
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Find f(8) given f(x) = 3/4x - 1.
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Solve and graph the inequality -3x < 10 - x.
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Is the ordered pair (2 , 1) a solution to 4x - 2y = 5?
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Find f(-4) given f(x) = x squared - 3.
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Solve and graph the inequality -4(3 - x) > 5(x + 1).
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Is the ordered pair (-2 , 2) a solution to 2 + 6x = -5y?
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Find g(-3) given G(x) = -x squared - 6.
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Identify the domain and range from the following points: (-2, 3), (4, 5), (-1, 6), (2, 3), (1, 3), (4, 7), (-3, 0).
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Simplify the expression 6n^2 - 5n^2.
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Is (-3,9) (9,-3) (7,0) (-3,6) a function?
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Simplify the expression -(-5 + 10x).
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Simplify the expression -4(p - (-5)).
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What is the x-intercept of the equation 4x - 3y = 12?
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Write an equation for the sentence 'The difference of 12w and -9 is -22' and solve for w.
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Solve for p in the equation -3z + 2w - p = -2z.
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Order the following numbers from least to greatest: -2 1/4, -7/4, 3/4, 4/3.
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Translate and solve: '4 less than 3 times a number is -16'.
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Find the x and y intercepts and graph for the equation 6x + 3y = 12.
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Find the x and y intercepts and graph for the equation -5y + 2x = -10.
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Rearrange the equation 2x - 3y = 9 into slope-intercept form.
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Rearrange the equation 4x = 2y - 8 into slope-intercept form.
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Study Notes
Algebraic Expressions and Definitions
- 6t: Represents "a number t times 6."
- 9n + 13: Represents "13 more than the product of 9 and n."
- Absolute value equations lead to multiple solutions (e.g., |2x - 4| = 22 yields x = -9 or 13).
- Functions: A relation where each input has exactly one output (e.g., (2, 1) is not a solution to 4x - 2y = 5 with inconsistent outputs).
Solving Equations
- Linear Equations: Solve equations like 4x + 12 = 2x - 6 gives x = -9.
- No Solutions: Some equations have no solution, such as 3(x - 3) = 3x - 6.
- Identity Equations: Equations like -2(x - 5) = 10 - 2x yield all real numbers as solutions.
- Finding Zero: An ordered pair (like (-2, 2)) is accepted if it satisfies the equation.
Simplification and Evaluation
- Simplification: Expressions like 36 ÷ (4 + 2^3) simplify to 3.
- Evaluation: Substitute values in expressions (e.g., f(8) = 5 for f(x) = (3/4)x - 1).
- Combine Like Terms: Expressions like 6n^2 - 5n^2 simplify to n^2.
Inequalities
- Graphing Solutions: Inequalities like -3x < 10 - x can be solved for x > -5 and graphed accordingly.
- Rejecting solutions: Solution pairs must fit the original equation conditions.
Rectangles and Geometry
- Perimeter Formulation: The perimeter equation 2L + 2w = 82 helps find dimensions when length is specified as 3 times the width plus one.
Intercepts and Functions
- Finding Intercepts: x-intercept of 4x - 3y = 12 is (3,0), while intercepts for linear equations can be computed similarly.
- Slope-Intercept Form: Rearranging equations like 2x - 3y = 9 to y = 2/3x - 3 presents the slope and y-intercept clearly.
Handling Negative and Fractional Values
- Negative Bases in Exponents: For (-1/5)^3, simplify to -1/125.
- Ordering Numbers: Arrange fractions and mixed numbers from least to greatest (e.g., -2 1/4, -7/4, 3/4, 4/3).
Problem-solving Techniques
- Translating Problems: Convert verbal math problems to algebraic equations (e.g., 4 less than 3 times a number is -16).
- Graphing Points: Solve equations to find coordinates for plotting graphs.
Absolute Values
- Absolute value equations, like |x - 3| - 6 = 2, yield multiple solutions by examining the distance from a point on the number line.
Identifying Domain and Range
- Domain and range can be determined from ordered pairs, capturing possible x and y values from relations.
Final Tips
- Ensure every algebraic expression is simplified correctly and all solutions are checked against original equations for validity.
- Practice using various methods to enhance problem-solving skills in algebra.
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Description
This quiz covers important concepts in algebra, including algebraic expressions, solving linear equations, and functions. It will test your understanding of simplification, evaluation, and the different types of equations. Prepare to explore a variety of problems that challenge your algebra skills!
