Algebraic Expressions, Linear Equations, and Inequalities: Precalculus Study Guide

Algebraic Expressions

Definitions and Identification

Algebraic expressions are fundamental in precalculus, representing combinations of numbers, variables, and operations. Understanding their structure is essential for simplifying and manipulating mathematical statements.

• Algebraic Expression: Any combination of numbers, variables, and arithmetic operations (addition, subtraction, multiplication, division).

• Term: Each part of an algebraic expression separated by addition or subtraction. For example, in $gt^2 - 2vt + 2s$, the terms are $gt^2$, $-2vt$, and $2s$.

• Factor: The quantities multiplied together within a term. In $-2vt$, the factors are $-2$, $v$, and $t$.

• Polynomial: An algebraic expression with only nonnegative integer exponents on one or more variables, and no variables in the denominator.

• Degree of a Term: The sum of the exponents of the variables in the term.

• Degree of a Polynomial: The highest degree among its terms.

• Monomial: A polynomial with one term.

• Binomial: A polynomial with two terms.

• Trinomial: A polynomial with three terms.

Example:

• For $4x^2 - 5x + 3$: Degree is 2; it is a trinomial.

• For $2x^6 - x$: Degree is 6; it is a binomial.

• For $3x$: Degree is 1; it is a monomial.

• For $xy^3 + 7x - 3$: Degree is 4 (from $xy^3$); it is a trinomial.

• For $6$: Degree is 0; it is a monomial.

Similar Terms and Simplification

Similar (or like) terms have identical variable parts and can be combined by addition or subtraction.

• Example: $8b - 6ab + 81b$; similar terms are $8b$ and $81b$.

• Simplification: Combine like terms and apply arithmetic operations.

Examples:

• $3x + 2x - 5y = 5x - 5y$

• $6a^2 - 7a + 8ax$ (no like terms)

• $6a + 5c + 2a - c = 8a + 4c$

Grouping Symbols and Parentheses

Expressions may include grouping symbols (parentheses, brackets, braces) that affect the order of operations.

• Example: $2(a + 2x) = 2a + 4x$

• Example: $-(a - 3c) = -a + 3c$

• Example: $3c + (2b - c) = 3c + 2b - c = 2b + 2c$

• Example: $3c - (2b - c) = 3c - 2b + c = -2b + 4c$

• Example: $3c - (-2b + c) = 3c + 2b - c = 2b + 2c$

• Example: $y(3 - y) - 2(y - 3) = 3y - y^2 - 2y + 6 = y - y^2 + 6$

Additional info:

• Nested parentheses require careful application of the distributive property and order of operations.

Multiplication of Algebraic Expressions

Multiplying Monomials and Polynomials

Multiplication of algebraic expressions involves multiplying coefficients and applying exponent rules.

• Product of Monomials: Multiply numerical coefficients and add exponents for like bases.

• Example: $3c^5 \times (-4c^2) = -12c^{7}$

• Example: $(-2b^2y^3) \times (-9ab y^5) = 18a b^3 y^8$

• Example: $2xy \times (-6c x^2) \times (3x c y^2) = 2xy \times -6c x^2 \times 3x c y^2 = -36 c^2 x^4 y^3$

Multiplying Powers of Monomials

• Example: $-3a \times (2a^2 x)^3 = -3a \times 8a^6 x^3 = -24a^7 x^3$

• Example: $2s^3 \times (-s t^4)^2 \times 4s^2 t = 2s^3 \times s^2 t^8 \times 4s^2 t = 8s^7 t^9$

Multiplying Monomials by Polynomials

• Example: $2a x (3a x^3 - 4y z) = 6a^2 x^4 - 8a x y z$

• Example: $5c y^2 (-7c x - a c) = -35c^2 x y^2 - 5a c^2 y^2$

Multiplying Polynomials

• Example: $(x - 2)(x + 3) = x^2 + 3x - 2x - 6 = x^2 + x - 6$

Polynomials Raised to a Power

• Example: $(x + 5)^2 = x^2 + 10x + 25$

• Example: $(2a - b)^3 = 8a^3 - 12a^2 b + 6a b^2 - b^3$

Division of Algebraic Expressions

Dividing Monomials and Polynomials

Division of algebraic expressions uses exponent rules and simplification techniques.

• Quotient of Monomials: Divide coefficients and subtract exponents for like bases.

• Example: $\frac{3c^7}{c^2} = 3c^{5}$

• Example: $\frac{16x^3 y^5}{4x y^2} = 4x^2 y^3$

• Example: $\frac{-6a^2 x y^2}{2a x y^4} = -3a y^{-2} = -3a \frac{1}{y^2}$

Dividing by a Monomial

• Example: $\frac{4a^2 + 8a}{2a} = 2a + 4$

• Example: $\frac{4x^3 y - 8x^3 y^2 + 2x^2}{2x^2 y} = 2x y - 4x y^2 + 1$

Division of One Polynomial by Another (Long Division)

Polynomial long division is analogous to numerical long division and follows a systematic process.

1. Arrange dividend and divisor in descending powers.

2. Divide the first term of the dividend by the first term of the divisor to get the first term of the quotient.

3. Multiply the divisor by this term and subtract from the dividend.

4. Repeat until the remainder is zero or of lower degree than the divisor.

5. Express the answer as: $\text{quotient} + \frac{\text{remainder}}{\text{divisor}}$

Example:

• Divide $6x^2 + x - 2$ by $2x - 1$ using long division.

• Divide $8x^3 + 4x^2 + 3$ by $4x^2 - 1$ using long division.

Solving Equations

Types and Solutions of Equations

Equations are statements asserting equality between two expressions. Solving equations involves finding values that satisfy the equality.

• Equation: A statement that two expressions are equal.

• Solving: Find values of the unknown variable that make the equation true, typically by performing the same operation on both sides.

Examples:

• $x - 3 = 12 \Rightarrow x = 15$

• $x + 3 = 12 \Rightarrow x = 9$

• $\frac{x}{3} = 12 \Rightarrow x = 36$

• $3x = 12 \Rightarrow x = 4$

• $2t - 7 = 9 \Rightarrow 2t = 16 \Rightarrow t = 8$

• $x - 7 = 3x - (6x - 8) \Rightarrow x - 7 = 3x - 6x + 8 \Rightarrow x - 7 = -3x + 8$

Procedure for Solving Equations

1. Remove grouping symbols (apply distributive law).

2. Combine like terms on each side.

3. Perform the same operations on both sides until the variable is isolated.

4. Check the solution in the original equation.

Classification of Equations

• Conditional Equation: Valid for only certain values of the unknown.

• Identity: Valid for all values of the unknown.

• Contradiction: Not valid for any values of the unknown.

Examples:

• $x^2 - 4 = (x - 2)(x + 2)$ is an identity.

• $x + 5 = x + 1$ is a contradiction (no solution).

Ratio and Proportion

• Ratio: $a/b$ is the ratio of $a$ to $b$.

• Proportion: An equation stating two ratios are equal.

• Example: Set up $\frac{x}{8} = \frac{3}{4}$ and solve for $x$.

Properties of Inequalities

Definitions and Types

Inequalities express relationships where one quantity is greater or less than another. Their solution sets can be conditional or absolute.

• Solution Set: The set of values that satisfy the inequality.

• Conditional Inequality: True for some, but not all, values of the variable.

• Absolute Inequality: True for all values of the variable.

Inequality Symbols and Translation

Symbol	Word Translation	Example
<	less than	$x < 5$
≤	less than or equal to	$x \le 5$
>	greater than	$x > 5$
≥	greater than or equal to	$x \ge 5$

Properties of Inequalities

Property	Symbolically	Example
Addition/Subtraction	If $a > b$, then $a + c > b + c$ and $a - c > b - c$	If $3 > 2$, then $3 + 1 > 2 + 1$
Multiplication/Division by Positive	If $a > b$, then $a c > b c$ and $a / c > b / c$, $c > 0$	If $2 > 1$, then $2 \times 3 > 1 \times 3$
Multiplication/Division by Negative	If $a > b$, then $a c < b c$ and $a / c < b / c$, $c < 0$	If $2 > 1$, then $2 \times (-1) < 1 \times (-1)$
Powers and Roots	If $a > b$, then $a^n > b^n$ and $\sqrt[n]{a} > \sqrt[n]{b}$, $n > 0$, $a, b > 0$	If $3 > 2$, then $3^2 > 2^2$

Solving Linear Inequalities

Single-Step and Multi-Step Inequalities

Solving linear inequalities follows similar steps to solving equations, but requires attention to the direction of the inequality, especially when multiplying or dividing by negative numbers.

• Example: $x + 2 < 4 \Rightarrow x < 2$

• Example: $\frac{x}{2} > 4 \Rightarrow x > 8$

• Example: $2x \le 4 \Rightarrow x \le 2$

• Example: $3 - 2x \ge 15 \Rightarrow -2x \ge 12 \Rightarrow x \le -6$

• Example: $2x \le 3 - x \Rightarrow 3x \le 3 \Rightarrow x \le 1$

• Example: $\frac{3}{2}(1 - x) > \frac{1}{4} - x$

Inequalities with Three Members

• Example: $-1 < 2x + 3 < 6$

• Example: $2x < x - 4 \le 3x + 8$

Interval Notation and Graphing

• Interval Notation: Used to represent solution sets, e.g., $x < 2$ is $(-\infty, 2)$.

• Graphing: Solutions can be represented on a number line, with open or closed circles indicating whether endpoints are included.

Summary Table: Types of Equations and Inequalities

Type	Definition	Example
Conditional Equation	True for some values	$x + 2 = 5$
Identity	True for all values	$x^2 - 4 = (x - 2)(x + 2)$
Contradiction	True for no values	$x + 5 = x + 1$
Conditional Inequality	True for some values	$x > 2$
Absolute Inequality	True for all values	$x^2 + 1 > 0$

Key Formulas and Properties

• Combining Like Terms: Add or subtract coefficients of terms with identical variable parts.

• Distributive Property: $a(b + c) = ab + ac$

• Exponent Rules: $x^m \times x^n = x^{m+n}$; $\frac{x^m}{x^n} = x^{m-n}$

• Solving Linear Equations: Isolate the variable using inverse operations.

• Solving Linear Inequalities: Apply operations, reverse inequality when multiplying/dividing by negative.

Practice and Applications

• Identify and classify algebraic expressions and polynomials.

• Simplify expressions using grouping symbols and combining like terms.

• Multiply and divide monomials and polynomials.

• Solve linear equations and inequalities, including those with three members.

• Apply properties of inequalities to solve and graph solution sets.

• Set up and solve ratio and proportion equations.

Additional info: These notes cover foundational algebraic concepts essential for success in precalculus, including manipulation of expressions, solving equations and inequalities, and understanding solution sets. Mastery of these topics is critical for progressing to more advanced topics such as functions, polynomials, and analytic geometry.