Review of Real Numbers and Order of Operations

Order of Operations (PEMDAS)

To evaluate mathematical expressions correctly, follow the order of operations:

• Parentheses

• Exponents

• Multiplication and Division (from left to right)

• Addition and Subtraction (from left to right)

Absolute value represents the distance from zero on the number line and is always non-negative.

Exponents indicate repeated multiplication of a base number.

Evaluating expressions means substituting values for variables and simplifying using the order of operations.

Example: Evaluate

• First, calculate inside parentheses:

• Absolute value:

• Subtract:

• Multiply:

• Add:

Linear Equations and Inequalities

Solving Linear Equations in One Variable

A linear equation in one variable can be written in the form .

• Isolate the variable using inverse operations.

• Check your solution by substituting back into the original equation.

Example: Solve

• Add 5 to both sides:

• Divide by 3:

Solving Formulas

To solve a formula for a specific variable, rearrange the equation to isolate that variable.

Example: Solve for

• Divide both sides by :

Applications of Linear Equations: Perimeter and Area

• Perimeter is the distance around a figure (units: ft, in, mi, etc.).

• Area is the measure of the surface (units: , , , etc.).

Use the 5-step problem-solving method:

1. Read and understand the problem.

2. Assign a variable.

3. Write an equation.

4. Solve the equation.

5. Check and interpret the solution.

Example: The perimeter of a rectangle is 24 ft. If the length is 7 ft, what is the width?

• Perimeter formula:

• Substitute:

• Solve: ft

Solving Inequalities

Solve inequalities similarly to equations, but reverse the inequality symbol when multiplying or dividing by a negative number.

• Graph solutions on a number line.

• Express solutions in set or interval notation.

Example: Solve

• Divide by -2 (reverse symbol):

Translating Inequalities

• At most means "less than or equal to" ().

• At least means "greater than or equal to" ().

Example: "No more than 8" translates to .

Graphing and Linear Equations in Two Variables

Solutions to Linear Equations in Two Variables

A linear equation in two variables, such as , has infinitely many solutions, each represented by an ordered pair .

Graphing Solutions

• Plot points that satisfy the equation.

• Draw a straight line through the points with arrowheads to indicate the line continues infinitely.

Finding Intercepts:

• x-intercept: Set and solve for .

• y-intercept: Set and solve for .

Slope of a Line

The slope measures the steepness of a line.

• Slope formula:

• Parallel lines have equal slopes.

• Perpendicular lines have slopes that are negative reciprocals.

• Zero slope: Horizontal line ().

• Undefined slope: Vertical line.

Slope-Intercept and Point-Slope Forms

• Slope-intercept form:

• Point-slope form:

Example: Write the equation of a line with slope 3 passing through (2, 5):

• Point-slope form:

Exponents and Polynomials

Exponent Rules

• (if )

Scientific Notation

Express numbers as , where and is an integer.

• To convert to scientific notation, move the decimal point so only one nonzero digit is to the left.

• To convert from scientific notation, move the decimal point places right (if ) or left (if ).

Example:

Polynomial Terminology

• Term: A number, variable, or product of numbers and variables.

• Coefficient: The numerical factor of a term.

• Degree: The highest exponent of the variable.

• Monomial: One term; Binomial: Two terms; Trinomial: Three terms.

Adding, Subtracting, and Multiplying Polynomials

• Add or subtract like terms (same variable and exponent).

• Multiply each term in one polynomial by each term in the other (distributive property).

Example:

Multiplying Special Products

• Square of a binomial:

• Difference of squares:

Dividing Polynomials by a Monomial

• Divide each term of the polynomial by the monomial.

Example:

Factoring Polynomials

Factoring Overview

Factoring means rewriting a polynomial as a product of simpler polynomials.

Common Factors and Grouping

• Factor out the greatest common factor (GCF) from all terms.

• For four-term polynomials, group terms to factor by grouping.

Example:

Factoring Trinomials

• Type : Find two numbers that multiply to and add to .

• Type : Use factoring by grouping or trial and error.

Example:

Example:

Factoring Special Forms

• Perfect square trinomial:

• Difference of squares:

Example:

Example:

General Mixed Factoring

Apply all factoring methods as appropriate to factor completely.

Solving Quadratic Equations by Factoring

• Set the equation to zero.

• Factor the quadratic expression.

• Set each factor equal to zero and solve for the variable.

Example: or

Applications: Consecutive Integers, Pythagorean Theorem, Area

• Consecutive integers: Let the first integer be , the next is , etc.

• Pythagorean theorem: For a right triangle,

• Area problems: Use area formulas and set up equations to solve for unknowns.

Systems of Linear Equations

Graphing Solutions to Systems

A system of linear equations consists of two or more equations. The solution is the point(s) where the graphs intersect.

• Consistent system: Has at least one solution.

• Inconsistent system: Has no solution (parallel lines).

• Dependent system: Equations represent the same line (infinitely many solutions).

Solving by Substitution

1. Solve one equation for one variable.

2. Substitute into the other equation.

3. Solve for the remaining variable.

4. Back-substitute to find the other variable.

Example: Solve and

• Substitute into the second equation:

• Then

• Solution:

Study Strategies

• Review each concept and practice problems you find challenging.

• Use previous exams and homework for targeted practice.

• Seek help from the Math Lab or discussion forums if needed.