Algebra Class: Expressions and Equations

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.