Openstax Intermediate Algebra 2E

In Lesson 1.1 of OpenStax Intermediate Algebra 2E, students review foundational algebraic concepts including factors, prime factorizations, least common multiples, variables, algebraic symbols, and the order of operations. The lesson also covers evaluating expressions, identifying and combining like terms, and translating English phrases into algebraic expressions. This chapter serves as a concise refresher of core algebra language skills needed throughout the Intermediate Algebra course.

In this lesson from OpenStax Intermediate Algebra 2E, Chapter 1, students learn to work with integers by simplifying expressions involving absolute value, adding, subtracting, multiplying, and dividing integers, and evaluating variable expressions with integers. Students also practice translating English phrases into algebraic expressions with integers and applying integer concepts to real-world problems. This foundational lesson builds essential skills for algebraic reasoning at the intermediate algebra level.

In Lesson 1.3 of OpenStax Intermediate Algebra 2E, Chapter 1: Foundations, students learn to simplify fractions using prime factorization and the Equivalent Fractions Property, then multiply, divide, add, and subtract fractions. The lesson also covers applying the order of operations to expressions involving fractions and evaluating variable expressions with fractions.

In this lesson from OpenStax Intermediate Algebra 2E, students review and strengthen core decimal skills including rounding to specified place values, adding and subtracting decimals by aligning decimal points, and multiplying and dividing decimals. The lesson also covers converting between decimals, fractions, and percents, simplifying square root expressions, and classifying numbers as integers, rational, irrational, or real. Students further practice locating fractions and decimals on the number line, building the foundational fluency needed for algebraic work throughout the course.

In this lesson from OpenStax Intermediate Algebra 2E, students learn how to solve linear equations using a general strategy, including determining whether a value is a solution by substitution, classifying equations, and solving equations with fraction or decimal coefficients. The lesson introduces the formal definition of a linear equation in one variable and walks through step-by-step simplification on both sides of an equation. This content is suited for intermediate algebra students developing foundational skills in algebraic reasoning.

In this lesson from OpenStax Intermediate Algebra 2E, students learn a structured seven-step problem-solving strategy for tackling algebraic word problems, including number problems, percent applications, and simple interest problems. The approach guides learners to read, identify, name with a variable, translate into an equation, solve, check, and answer each problem systematically. This skill is foundational for intermediate algebra students who need a reliable method to convert real-world situations into linear equations.

In this lesson from OpenStax Intermediate Algebra 2E, students learn to solve coin word problems, ticket and stamp word problems, mixture word problems, and uniform motion applications using linear equations. The core technique involves setting up algebraic models — such as number × value = total value — to translate real-world scenarios into solvable equations. This intermediate algebra lesson builds practical problem-solving skills applicable across a range of multi-variable word problems.

In this lesson from OpenStax Intermediate Algebra 2E, students learn to graph linear inequalities on the number line and express solution sets using interval notation, including open and closed endpoints represented by parentheses and brackets. The lesson covers solving linear inequalities, translating word problems into inequality statements, and applying these skills to real-world scenarios. This intermediate algebra content builds foundational skills for working with solution sets, infinity symbols, and compound inequalities.

In this lesson from OpenStax Intermediate Algebra 2E, students learn to solve absolute value equations and inequalities by isolating the absolute value expression and rewriting it as two equivalent equations or compound inequalities. The lesson covers cases involving "less than" and "greater than" absolute value inequalities, including how to express solution sets on a number line and in interval notation. Real-world applications of absolute value are also explored, making this a key topic in intermediate algebra for high school students.

In this lesson from OpenStax Intermediate Algebra 2E, students learn how to plot ordered pairs on a rectangular coordinate system and graph linear equations in two variables by plotting points. The lesson covers key concepts including quadrants, the origin, x- and y-intercepts, and how to graph vertical, horizontal, and sloped lines using intercepts. It is designed for intermediate algebra students building foundational skills in coordinate geometry and linear relationships.

In this lesson from OpenStax Intermediate Algebra 2E, students learn how to find and interpret the slope of a line using the rise-over-run formula, including calculating slope from a graph, identifying zero slope for horizontal lines, and recognizing undefined slope for vertical lines. The lesson also covers graphing lines using slope and y-intercept, and using slope to identify parallel and perpendicular lines. Real-world applications such as roof pitch and road grades help students connect the concept of slope as a rate of change to everyday contexts.

In this lesson from OpenStax Intermediate Algebra 2E, students learn how to identify relations as sets of ordered pairs and find their domain and range using ordered pairs, mapping diagrams, and graphs. Students also explore how to determine whether a relation qualifies as a function and practice evaluating functions for specific values. The lesson builds foundational skills in function notation and analysis central to algebra.

This lesson from OpenStax Intermediate Algebra 2E introduces students to systems of linear equations with two variables, teaching them how to determine whether an ordered pair is a solution and how to solve systems using graphing, substitution, and elimination. Students learn to identify consistent, inconsistent, and dependent systems and practice choosing the most efficient method for a given problem. The material builds directly on prior knowledge of linear equations and slope-intercept form.

In this lesson from OpenStax Intermediate Algebra 2E, students learn to solve real-world word problems by translating them into systems of linear equations and applying methods such as substitution to find solutions. Topics covered include direct translation applications, geometry problems, and uniform motion scenarios involving two variables. Students follow a structured seven-step problem-solving process to set up, solve, and verify systems of equations in practical contexts.

In this lesson from Openstax Intermediate Algebra 2E, students learn how to solve mixture applications using systems of linear equations, including ticket and coin problems, interest calculations, and cost and revenue functions. Students practice setting up two-variable systems from organized data tables and applying the elimination and substitution methods to find unknown quantities. The lesson is suitable for intermediate algebra students working through Chapter 4 on systems of linear equations.

In this lesson from OpenStax Intermediate Algebra 2E, students learn how to solve systems of linear equations with three variables by working with ordered triples of the form (x, y, z) and applying elimination across three equations. The lesson covers how to determine whether an ordered triple is a solution, solve three-variable systems algebraically, and interpret the geometric relationship between planes in three-dimensional space. Real-world applications using three-variable systems are also introduced, making this an important extension of earlier two-variable equation work.

In this lesson from OpenStax Intermediate Algebra 2E, students learn how to evaluate the determinant of 2×2 and 3×3 matrices and apply Cramer's Rule to solve systems of linear equations. The lesson introduces key concepts including square matrices, minors, and expansion by minors, building the algebraic foundation needed to use determinants as a solution method. This content is appropriate for intermediate algebra students at the high school or early college level.

Students learn how to graph systems of linear inequalities on a coordinate plane, identify the overlapping shaded region that represents all solutions, and determine whether ordered pairs satisfy a system of two or more inequalities. The lesson covers writing boundary lines as solid or dashed based on the inequality symbol and testing points to confirm solution regions. This content is from Chapter 4 of OpenStax Intermediate Algebra 2E, part of a unit on systems of linear equations and inequalities.

In this lesson from OpenStax Intermediate Algebra 2E, students learn to identify and classify polynomials as monomials, binomials, or trinomials, and determine the degree of a polynomial by analyzing the exponents of its terms. The lesson then guides students through adding and subtracting polynomials, including evaluating and performing operations on polynomial functions. These foundational skills support further work with polynomial multiplication, division, and real-world applications covered throughout Chapter 5.

In this lesson from OpenStax Intermediate Algebra 2E, students learn how to find the greatest common factor (GCF) of two or more polynomial expressions by factoring coefficients into primes and identifying shared variables. They then apply the Distributive Property in reverse to factor the GCF out of a polynomial and use the factor by grouping method to factor expressions with four or more terms. These foundational factoring skills build the algebraic fluency needed for solving polynomial equations throughout Chapter 6.

In this lesson from OpenStax Intermediate Algebra 2E, students learn how to apply a general step-by-step strategy for factoring polynomials completely, choosing among methods such as greatest common factor, difference of squares, sum or difference of cubes, trinomial factoring, and grouping based on the structure of the polynomial. The lesson emphasizes recognizing whether a polynomial is a binomial, trinomial, or has more than three terms before selecting the correct technique. It is part of Chapter 6 and is appropriate for intermediate algebra students at the high school or early college level.

In this lesson from OpenStax Intermediate Algebra 2E, students learn to solve polynomial equations of degree two — known as quadratic equations — using the Zero Product Property and factoring techniques. The lesson covers writing equations in standard form ax² + bx + c = 0, factoring the polynomial expression, and applying the Zero Product Property to find solutions. Students also extend these skills to solve equations involving polynomial functions and real-world applications modeled by polynomial equations.

In this lesson from OpenStax Intermediate Algebra 2E, students learn to work with rational expressions — fractions whose numerators and denominators are polynomials. The lesson covers how to determine values that make a rational expression undefined, simplify rational expressions by factoring, and multiply and divide rational expressions and rational functions. These skills build directly on fraction operations and prepare students to solve rational equations and real-world applications later in Chapter 7.

In this lesson from OpenStax Intermediate Algebra 2E, students learn to simplify complex rational expressions — rational expressions that contain other rational expressions in their numerator or denominator. The lesson covers two methods: rewriting the complex expression as division and multiplying by the reciprocal, and using the least common denominator (LCD) to combine and reduce terms. Students practice applying factoring, finding LCDs, and identifying excluded values throughout the simplification process.

In this lesson from OpenStax Intermediate Algebra 2e, students learn how to solve rational equations by multiplying both sides by the least common denominator (LCD) to clear fractions, then solving the resulting equation. A key focus is identifying and discarding extraneous solutions — values that make a denominator equal to zero and would render the original expression undefined. Students also practice applying this method to rational functions and solving rational equations for a specific variable.

In this intermediate algebra lesson from OpenStax Intermediate Algebra 2E, students learn to solve real-world applications using rational equations, covering proportions, similar figures, uniform motion, work problems, and direct and inverse variation. Students practice setting up and solving equations of the form a/b = c/d by clearing fractions with the LCD or using cross-multiplication, then applying these techniques to practical scenarios such as medication dosing and distance-rate-time problems. This lesson is suitable for high school students working through rational expressions and functions.

In OpenStax Intermediate Algebra 2E, Lesson 8.1 introduces students to simplifying expressions with roots, covering square roots, cube roots, and nth roots, including the concepts of principal square root, radical sign, and radicand. Students learn to simplify both numeric and variable radical expressions, estimate and approximate roots, and recognize when a root is not a real number. This lesson serves as the foundation for the broader chapter on roots and radicals.

In this lesson from OpenStax Intermediate Algebra 2E, students learn how to simplify radical expressions using the Product Property and Quotient Property of nth Roots. The lesson covers identifying perfect square, cube, and fourth power factors within a radicand and rewriting expressions such as the square root of 98 as 7 times the square root of 2. This material is part of Chapter 8 on Roots and Radicals, typically studied at the intermediate algebra level in high school or early college coursework.

In this lesson from OpenStax Intermediate Algebra 2E, students learn to simplify expressions with rational exponents of the form a^(1/n) and a^(m/n) by connecting them to radical notation, where the denominator of the exponent corresponds to the index of the radical. Students practice converting between rational exponent and radical forms and applying the properties of exponents — including the Power Property and negative exponent rule — to simplify numerical and algebraic expressions. This intermediate algebra topic builds on prior knowledge of integer exponents and nth roots.

In this lesson from OpenStax Intermediate Algebra 2E, students learn how to divide radical expressions using the Quotient Property of Radical Expressions, simplifying fractions involving square roots and cube roots with variable radicands. The lesson also introduces rationalizing the denominator, a technique for rewriting fractions so no radical remains in the denominator, including both one-term and two-term denominators. This content is appropriate for intermediate algebra students and builds on prior skills with exponent rules and polynomial multiplication.

In this lesson from Openstax Intermediate Algebra 2E, students learn how to solve radical equations by isolating the radical and raising both sides of the equation to the power of the index to eliminate it. The lesson covers solving equations with one or two radicals, identifying extraneous solutions, and handling cases where squaring a binomial requires the Product of Binomial Squares Pattern. Students also apply these techniques to real-world problems involving radical expressions.

In this Openstax Intermediate Algebra 2E lesson on the complex number system, students learn to evaluate square roots of negative numbers using the imaginary unit i, where i² = −1, and to write results in standard form a + bi. The lesson covers adding, subtracting, multiplying, and dividing complex numbers, as well as simplifying powers of i. This chapter-closing section bridges real number concepts students already know with the broader complex number system used in higher mathematics.

In this lesson from OpenStax Intermediate Algebra 2E, students learn to solve quadratic equations using the Square Root Property, applying it to equations of the form ax² = k and a(x−h)² = k. The lesson covers how to isolate the squared term and take both the positive and negative square roots to find two solutions, including cases where the result is an irrational number left in radical form. This foundational skill prepares students for more advanced methods such as completing the square and the quadratic formula covered later in Chapter 9.

This lesson from Openstax Intermediate Algebra 2E teaches students how to solve quadratic equations by completing the square, covering both the forms x² + bx + c = 0 and ax² + bx + c = 0. Students learn to recognize and apply the binomial squares pattern to transform expressions into perfect square trinomials, then apply the Square Root Property to find solutions. The technique builds on prior skills in factoring and expanding binomials and lays groundwork for later work with conic sections.

Students learn to solve quadratic equations of the form ax² + bx + c = 0 using the Quadratic Formula, x = (−b ± √(b² − 4ac)) / 2a, derived by completing the square on the general standard form. The lesson also covers using the discriminant (b² − 4ac) to predict the number and type of solutions, and choosing the most appropriate method for solving a given quadratic equation. This material is from Chapter 9 of the OpenStax Intermediate Algebra 2e textbook.

In this lesson from OpenStax Intermediate Algebra 2E, Chapter 9, students learn how to solve equations in quadratic form by using u-substitution to rewrite higher-degree or complex equations, such as fourth-degree polynomials like 6x⁴ − 7x² + 2 = 0, into the standard ax² + bx + c = 0 form. Students practice identifying the appropriate substitution variable, solving the resulting quadratic equation by factoring or the Square Root Property, and back-substituting to find all solutions for the original variable.

In this lesson from OpenStax Intermediate Algebra 2E, students learn to solve real-world applications modeled by quadratic equations using all four methods: factoring, the Square Root Property, completing the square, and the Quadratic Formula. The lesson covers problem types including consecutive odd and even integers, where students set up and solve equations such as n(n+2) = 195 using a structured seven-step problem-solving strategy. Students practice selecting the most efficient solution method for each application and interpreting results in context.

In this lesson from OpenStax Intermediate Algebra 2E, students learn to graph quadratic functions using transformations, including vertical shifts of the form f(x) = x² + k, horizontal shifts of the form f(x) = (x − h)², and vertical stretches and compressions of the form f(x) = ax². Students also practice combining these transformations and working backwards from a graph to identify the quadratic function it represents.

In Lesson 9.8 of OpenStax Intermediate Algebra 2E, students learn to solve quadratic inequalities in standard form both graphically and algebraically. The graphical method involves identifying where a parabola lies above or below the x-axis, while the algebraic method builds on techniques from solving linear, rational, and quadratic equations. Solutions are expressed in interval notation.

In this lesson from OpenStax Intermediate Algebra 2E, students learn how to find and evaluate composite functions using the notation (f ∘ g)(x) = f(g(x)), determine whether a function is one-to-one, and find the inverse of a function. The lesson covers how composition differs from multiplication of functions and walks through step-by-step examples with linear expressions. This content introduces the foundational function concepts needed before exploring exponential and logarithmic functions in Chapter 10.

In this lesson from OpenStax Intermediate Algebra 2E, students learn to define and graph exponential functions of the form f(x) = a^x, exploring key properties such as domain, range, horizontal asymptotes, and characteristic points like (0, 1) and (1, a). Learners compare exponential growth curves for bases greater than 1, solve exponential equations, and apply these models to real-world contexts such as population growth, radioactive decay, and compound interest.

In this lesson from OpenStax Intermediate Algebra 2E, students learn to convert between exponential and logarithmic form, evaluate logarithmic functions by solving equations such as log_x 36 = 2, and graph logarithmic functions as the inverse of exponential functions. The lesson introduces the logarithmic function f(x) = log_a x with base a, explaining why logarithms are defined as the inverse of exponential functions. Students also solve logarithmic equations and explore real-world applications using logarithmic models.

In this lesson from OpenStax Intermediate Algebra 2E, students learn to apply the core properties of logarithms — including the Product Property, inverse properties, and the Change of Base Formula — to evaluate and simplify logarithmic expressions. Students practice rewriting expressions such as log_a(M·N) as a sum of logarithms and using inverse relationships like log_a(a^x) = x to solve problems efficiently. This lesson builds essential algebraic skills for solving exponential and logarithmic equations at the intermediate algebra level.

In this lesson from OpenStax Intermediate Algebra 2E, students learn to solve logarithmic equations using the One-to-One Property, the Product Property, the Quotient Property, and the Power Property of logarithms. They also practice solving exponential equations by applying common logarithms or natural logarithms when both sides cannot be written with the same base. Real-world exponential models are included as applications, making this an essential section for intermediate algebra students mastering transcendental equations.

In this lesson from OpenStax Intermediate Algebra 2E, students learn to apply the Distance Formula and Midpoint Formula to find lengths and midpoints between coordinate pairs on the rectangular coordinate system. The lesson also covers writing the standard form equation of a circle and graphing circles, connecting the Pythagorean Theorem to the geometry of conic sections. These foundational skills introduce Chapter 11's broader study of conics, including parabolas, ellipses, and hyperbolas.

In this lesson from Openstax Intermediate Algebra 2E, Chapter 11, students learn to graph and write equations of ellipses in standard form, including identifying the major axis, minor axis, vertices, foci, and intercepts. Students practice working with the standard form equation x²/a² + y²/b² = 1 for ellipses centered at the origin, as well as ellipses with centers shifted away from the origin using the translation form (x−h)²/a² + (y−k)²/b² = 1. The lesson also covers real-world applications involving ellipses.

In this lesson from OpenStax Intermediate Algebra 2E, students learn how to solve systems of nonlinear equations — where at least one equation is not linear — using three methods: graphing, substitution, and elimination. Learners practice finding ordered pair solutions at the intersection of curves such as parabolas, circles, and hyperbolas, and apply these techniques to real-world application problems.

In this lesson from OpenStax Intermediate Algebra 2E, students learn the definition of a sequence as a function whose domain is the counting numbers, and practice writing the first few terms using a general term formula (nth term notation such as aₙ = 2n). The lesson covers distinguishing between finite and infinite sequences, applying factorial notation, finding partial sums, and using summation notation to express a series compactly. This content is appropriate for intermediate algebra students and lays the foundation for arithmetic sequences, geometric sequences, and the Binomial Theorem covered later in Chapter 12.

In this lesson from OpenStax Intermediate Algebra 2E, students learn how to identify arithmetic sequences by finding the common difference between consecutive terms, write the general (nth) term formula, and calculate the sum of the first n terms of an arithmetic sequence. The lesson builds algebraic reasoning skills through worked examples, guided practice, and real-world applications of linear patterns.

In this lesson from OpenStax Intermediate Algebra 2E, students learn to identify geometric sequences by finding the common ratio between consecutive terms, derive the general nth term formula, and calculate finite and infinite geometric series sums. The lesson also covers real-world applications of geometric sequences and series. It is part of Chapter 12 and is suited for intermediate algebra students at the high school or early college level.

This lesson from OpenStax Intermediate Algebra 2e covers the Binomial Theorem and its applications, teaching students how to use Pascal's Triangle to find coefficients and expand binomial expressions such as (a + b)^n for any power. Students learn to identify patterns in binomial expansions, including how exponents shift across terms and how coefficients are derived using both Pascal's Triangle and binomial coefficients. The lesson also introduces formal notation for evaluating binomial coefficients as part of a broader study of sequences and series.