Algebra Basics and Operations

Basic Operations and Simplifications

• Simplifying expressions often involves combining like terms, e.g., 6(-5) simplifies to -30.
• Adding negative numbers requires careful arithmetic, e.g., -54 + 18 results in -36.
• The multiplicative inverse of a fraction like 1/2 is found by flipping it, giving a result of 2.

Evaluating Expressions

• Performing operations with both addition and subtraction can change results significantly, e.g., 15 - (-11) equals 26.
• The value of a variable raised to a power can be evaluated, as A^2 for A = -3 is 9.
• When simplifying sums of negative and positive values, such as (-5) + (-4) + (7), the result is -2.

Variable Operations

• In algebraic expressions, factorization helps simplify products, e.g., 6r · s · 4rt simplifies to 24r²st.
• When multiplying variables, like (-2/3 x)(3y)(-2x), the coefficient can be extracted, giving a result of 4.

Identifying Terms

• Counting terms in polynomial expressions is straightforward, e.g., the expression x² - 10xy + 3y + y² - 1 contains 5 terms.

Absolute Values and Negatives

• Absolute values influence simplification; for instance, -|-5 + 2| simplifies to -3.
• Evaluating mixed arithmetic expressions can vary widely, demonstrated by 6 - (-2) - 3(-5) simplifying to 23.

Integer Properties

• Integers are classified as never irrational numbers, showcasing their clear numerical nature.

Equivalency and Comparison

• Recognizing equivalent expressions is crucial; e.g., 0.25 can be represented as (0.5)², |-3 + 2.75|, or -(-0.25).
• When ordering fractions and decimals, (1/2)², 0.45, and 2/3 can be sequenced from smallest to largest.

Expression Simplification Techniques

• Collecting like terms and simplifying linear expressions is key; simplifying 5m - 13n - n + 4m results in 9m - 14n.
• The order of operations in arithmetic affects the evaluation; for 21 - 18 ÷ 3 · 2, the result is 9.

Advanced Evaluations

• Substitution into expressions reveals values, e.g., x²/y + x²/z with x = 10, y = -5, and z = -2 evaluates to -70.
• Plugging in values for x allows for straightforward evaluations, e.g., 7 + (-3x²) when x = 0 results in 7.

Additional Operation Handling

• Polynomial forms can be transformed: simplifying 7(x + 3) gives 7x + 21, illustrating distribution.
• Identifying equivalent expressions aids in simplifying, for instance, 10q - 20 can be factored as 10(q - 2).

Expression Construction

• Formulating expressions based on descriptions helps in problem-solving, e.g., "the difference between five times a number and twice that number" translates to 5N - 2N.

Final Simplifications

• Addition and subtraction within expressions such as 6x + 6 + 5x - 8 can be simplified to 11x - 2.
• Expression distribution is emphasized with 3(x - 4) + 8(x + 2) simplifying to 11x + 4.