Chapter 2: Equations and Inequalities

2.1 Solving Linear Equations in One Variable (Including Fractions)

Linear equations in one variable are equations that can be written in the form ax + b = c, where a, b, and c are constants. Solving these equations involves isolating the variable on one side of the equation.

• Key Point 1: To solve equations with fractions, multiply both sides by the least common denominator (LCD) to clear fractions.

• Key Point 2: Some equations may have no solution (inconsistent equations) or all real numbers as solutions (identities).

• Key Point 3: Always check your solution by substituting back into the original equation.

• Example: Solve Step 1: Multiply both sides by 6 (LCD): Step 2: Subtract 3x from both sides: Step 3: Add 1 to both sides:

2.4 Solving Linear Inequalities in One Variable (Including Fractions)

Linear inequalities are similar to linear equations but use inequality symbols (<, >, ≤, ≥). The solution is often a range of values, represented on a number line and in interval notation.

• Key Point 1: To solve inequalities with fractions, clear denominators as with equations.

• Key Point 2: If you multiply or divide both sides by a negative number, reverse the inequality sign.

• Key Point 3: Some inequalities have no solution or all real numbers as solutions.

• Key Point 4: Express solutions in interval notation and graph them on a number line.

• Example: Solve Step 1: Add 3 to both sides: Step 2: Multiply both sides by 2: Interval Notation:

2.5 Compound Linear Inequalities (Double/And/Or)

Compound inequalities involve two inequalities joined by "and" or "or". The solution is the intersection (for "and") or union (for "or") of the individual solutions.

• Key Point 1: "And" inequalities (e.g., ) require values that satisfy both inequalities.

• Key Point 2: "Or" inequalities (e.g., or ) require values that satisfy at least one inequality.

• Key Point 3: Graph solutions and write them in interval notation.

• Example: Solve Step 1: Subtract 1 from all parts: Step 2: Divide all parts by 3: Interval Notation:

2.3 Solving for a Specified Variable (Literal Equations)

Literal equations involve solving for one variable in terms of others. This is common in formulas from geometry, physics, and other sciences.

• Key Point 1: Use inverse operations to isolate the specified variable.

• Key Point 2: Rearranging formulas is a valuable skill for solving real-world problems.

• Example: Solve for y: Step 1: Subtract from both sides: Step 2: Divide both sides by 3:

Chapter 3: Graphs and Functions

3.1 Linear and Nonlinear Equations

Equations can be classified as linear or nonlinear based on the degree and arrangement of variables.

• Key Point 1: Linear equations have variables to the first power and graph as straight lines (e.g., ).

• Key Point 2: Nonlinear equations include powers other than one, products of variables, or variables in denominators (e.g., , ).

• Key Point 3: The graph of a linear equation is a straight line; nonlinear equations produce curves such as parabolas, circles, or hyperbolas.

• Example: Linear: (straight line) Nonlinear: (parabola)

3.2 Functions: Identification, Domain, Range, and Function Notation

A function is a relation in which each input (x-value) has exactly one output (y-value). Functions can be represented by sets of ordered pairs, equations, or graphs.

• Key Point 1: To determine if a relation is a function, check that no input value is paired with more than one output.

• Key Point 2: The domain is the set of all possible input values (x-values); the range is the set of all possible output values (y-values).

• Key Point 3: Function notation is written as , which means "the value of the function f at x".

• Key Point 4: To evaluate a function, substitute the given value for x into the function rule.

• Example 1 (Ordered Pairs): Is the relation a function? Answer: No, because the input 1 is paired with two different outputs (2 and 5).

• Example 2 (Domain and Range): For , domain is all real numbers; range is .

• Example 3 (Function Notation): If , find :

Table: Linear vs. Nonlinear Equations

Additional info: This guide covers the main topics and skills assessed on the test, including solving equations and inequalities (with fractions), compound inequalities, literal equations, distinguishing linear and nonlinear equations, and foundational concepts of functions, domain, range, and function notation.