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College Algebra
Unit 2 Overview: Equations and Inequalities
2.1 The Rectangular Coordinate Systems and Graphs
2.2 Linear Equations in One Variable
2.3 Models and Applications
2.4 Complex Numbers
2.5 Quadratic Equations
2.6 Other Types of Equations
2.7 Linear Inequalities and Absolute Value Inequalities

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2.6 Other Types of Equations

Last Updated on June 24, 2024

Equations with special properties require unique solving techniques. From rational exponents to absolute values, each type demands a specific approach. Understanding these methods is crucial for tackling complex algebraic problems effectively.

Logarithmic and exponential equations add another layer of complexity. By mastering logarithm properties and exponential rules, you'll be equipped to solve these equations confidently. Remember, choosing the right method is key to efficient problem-solving.

Solving Equations with Special Properties

Rational exponents and radicals

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  • Rational exponents
    • Defined as amn=amna^{\frac{m}{n}} = \sqrt[n]{a^m} expresses roots and powers together
    • Solve by raising both sides to the reciprocal exponent (nm\frac{n}{m}) to isolate the variable
  • Radical equations
    • Isolate radical term on one side of equation (x+1=3\sqrt{x+1} = 3)
    • Raise both sides to the power of the radical's index to eliminate it ((x+1)2=32(\sqrt{x+1})^2 = 3^2)
    • Simplify and solve resulting equation (x+1=9x+1 = 9, so x=8x = 8)
    • Check solutions in original equation to eliminate extraneous ones
    • Consider domain restrictions to ensure valid solutions

Factoring for polynomial equations

  • Factor polynomial to find its roots
    • Common factors: Factor out greatest common factor (GCF) (6x2+18x=6x(x+3)6x^2 + 18x = 6x(x+3))
    • Difference of squares: a2−b2=(a+b)(a−b)a^2 - b^2 = (a+b)(a-b) (x2−25=(x+5)(x−5)x^2 - 25 = (x+5)(x-5))
    • Sum or difference of cubes: a3±b3=(a±b)(a2∓ab+b2)a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2) (x3−8=(x−2)(x2+2x+4)x^3 - 8 = (x-2)(x^2 + 2x + 4))
    • Trinomials: ax2+bx+c=(px+q)(rx+s)ax^2 + bx + c = (px + q)(rx + s), where p+r=bp + r = b and pr=acpr = ac (2x2+7x+3=(2x+1)(x+3)2x^2 + 7x + 3 = (2x+1)(x+3))
  • Set each factor to zero and solve for variable to find roots
  • Solutions are roots that make polynomial equal zero

Absolute value equation solutions

  • Isolate absolute value expression on one side (∣2x−3∣=7|2x-3| = 7)
  • Consider two cases: expression inside is positive or negative
    1. ∣x∣=a|x| = a is equivalent to x=ax = a or x=−ax = -a (∣2x−3∣=7|2x-3| = 7 becomes 2x−3=72x-3 = 7 or 2x−3=−72x-3 = -7)
    2. ∣x∣=−a|x| = -a has no solution for a>0a > 0 (∣2x−3∣=−7|2x-3| = -7 has no solution)
  • Solve each case and combine solutions (x=5x = 5 or x=−2x = -2 for ∣2x−3∣=7|2x-3| = 7)
  • Interpret solutions based on problem context

Solving Equations with Logarithms and Exponentials

Logarithmic and exponential equations

  • Logarithmic equations
    • Use logarithm properties to simplify
      • Product property: log⁡b(MN)=log⁡b(M)+log⁡b(N)\log_b(MN) = \log_b(M) + \log_b(N)
      • Quotient property: log⁡b(MN)=log⁡b(M)−log⁡b(N)\log_b(\frac{M}{N}) = \log_b(M) - \log_b(N)
      • Power property: log⁡b(Mn)=nlog⁡b(M)\log_b(M^n) = n\log_b(M)
    • Isolate logarithm on one side (log⁡2(x)+log⁡2(x−1)=3\log_2(x) + \log_2(x-1) = 3)
    • Apply inverse function (exponential) to both sides to solve (23=x(x−1)2^3 = x(x-1))
  • Exponential equations
    • Isolate exponential expression on one side (3e2x=753e^{2x} = 75)
    • Apply inverse function (logarithm) to solve (ln⁡(3e2x)=ln⁡(75)\ln(3e^{2x}) = \ln(75), so 2x=ln⁡(25)2x = \ln(25))
    • If bases differ, use change of base formula: log⁡b(x)=log⁡a(x)log⁡a(b)\log_b(x) = \frac{\log_a(x)}{\log_a(b)}

Multiple solution methods

  • Recognize equations solvable by multiple methods
    • Factoring, quadratic formula, completing square for quadratics
    • Logarithms and exponentials for equations with those forms
    • Absolute value equations breakable into two cases
  • Choose most efficient method based on equation structure
    • Factoring for factorable quadratics (x2+5x+6=0x^2 + 5x + 6 = 0)
    • Quadratic formula for unfactorable quadratics (2x2−3x−2=02x^2 - 3x - 2 = 0)
    • Logarithms for exponential equations (5e3x=205e^{3x} = 20)
  • Verify solutions in original equation to check work

Advanced Equation Solving Techniques

Equation types and solution strategies

  • Algebraic manipulation: Rearranging terms and applying operations to isolate variables
  • Substitution method: Replacing variables with expressions to simplify complex equations
  • Graphical solutions: Using graphs to visualize and find intersection points of functions
  • Inverse functions: Applying inverse operations to both sides of an equation to solve for a variable

Key Terms to Review (32)

Completing the square: Completing the square is a method to solve quadratic equations by converting them into a perfect square trinomial. This facilitates easier solving and helps in deriving the quadratic formula.
Difference of squares: The difference of squares is a specific type of polynomial that takes the form $a^2 - b^2$, which can be factored into $(a + b)(a - b)$. It is based on the property that the product of a sum and a difference of two terms results in the difference of their squares.
Extraneous solutions: Extraneous solutions are solutions derived from the algebraic manipulation of an equation that do not satisfy the original equation. They often arise when both sides of an equation are squared or when absolute value functions are involved.
Factoring: Factoring is the process of breaking down an expression into a product of simpler expressions, often polynomials. It simplifies solving equations by expressing them as a product of factors.
Polynomial equation: A polynomial equation is an equation involving a polynomial expression, which sets the polynomial equal to zero. It can have one or more variables with non-negative integer exponents.
Quadratic formula: The quadratic formula is used to find the roots of a quadratic equation of the form $ax^2 + bx + c = 0$. It is given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
Radical equation: A radical equation is an equation in which the unknown variable appears under a radical sign, most commonly a square root. Solving these equations typically involves isolating the radical expression and then squaring both sides to eliminate the radical.
Substitution method: The substitution method is a technique for solving systems of equations by substituting one equation into another. This transforms the system into a single-variable equation that can be solved more easily.
Absolute Value Equations: An absolute value equation is a mathematical equation that involves the absolute value function, which represents the distance of a number from zero on the number line. These equations are used to solve for the unknown value that satisfies the given absolute value condition.
Algebraic Manipulation: Algebraic manipulation refers to the process of performing various operations and transformations on algebraic expressions to simplify, solve, or rearrange them. It involves the strategic use of mathematical rules and properties to manipulate expressions and equations in a logical and systematic manner.
Common Factors: Common factors are the numbers or variables that are shared between two or more expressions or equations. They represent the elements that are present in multiple mathematical expressions and can be factored out to simplify the expressions or solve the equations.
Completing the Square: Completing the square is a technique used to solve quadratic equations and transform quadratic functions into a more useful form. It involves rearranging a quadratic expression into a perfect square plus or minus a constant, allowing for easier analysis and manipulation of the equation or function.
Change of Base Formula: The change of base formula is a mathematical expression that allows for the conversion of logarithms from one base to another. This formula is particularly important in the context of logarithmic functions, their graphs, and the properties and equations involving logarithms and exponentials.
Difference of Squares: The difference of squares is a special type of polynomial expression where the terms are the difference between two perfect squares. This concept is particularly important in the context of factoring polynomials, working with rational expressions, solving quadratic equations, and understanding the properties of power functions and polynomial functions.
Difference of Cubes: The difference of cubes is a special case of polynomial factorization where a polynomial expression can be factored by recognizing the difference between two cubes. This factorization technique is useful in solving certain types of equations and understanding the behavior of power functions and polynomial functions.
Domain Restrictions: Domain restrictions refer to the limitations or constraints placed on the set of input values or independent variables that a function or equation can accept. These restrictions define the valid range of values for which the function or equation is defined and can be meaningfully evaluated.
Exponential Equations: Exponential equations are mathematical expressions where the unknown variable appears as the exponent. These equations describe situations where a quantity grows or decays at a constant rate over time, and they are commonly used to model real-world phenomena such as population growth, radioactive decay, and compound interest.
Extraneous Solutions: Extraneous solutions are solutions to an equation that do not satisfy the original conditions or restrictions of the problem. They are solutions that are introduced during the process of solving the equation but do not belong to the set of valid solutions for the given problem.
Equation Types: Equation types refer to the different classifications or categories of mathematical equations based on their structure, variables, and the relationships between the components. These equation types are crucial in the study of college algebra, as they determine the appropriate methods and strategies for solving and analyzing various types of equations.
Graphical Solutions: Graphical solutions refer to the process of solving equations or systems of equations by using graphs and visual representations. This approach allows for a more intuitive understanding of the relationships between variables and the behavior of the equations.
Factoring: Factoring is the process of breaking down a polynomial or algebraic expression into a product of smaller, simpler expressions. It involves identifying common factors and using various techniques to express a polynomial as a product of its factors. Factoring is a fundamental algebraic skill that is essential for understanding and manipulating polynomials, rational expressions, quadratic equations, and other types of equations and functions.
Inverse Functions: An inverse function is a function that undoes the operation of another function. It reverses the relationship between the input and output values, allowing you to find the original input when given the output. Inverse functions are particularly important in the study of exponential and other types of equations.
Logarithmic Equations: A logarithmic equation is an equation in which the unknown variable appears as the exponent of a logarithmic function. These equations are used to model and solve problems involving exponential growth and decay, as well as other applications where the relationship between variables is logarithmic in nature.
Quadratic Formula: The quadratic formula is a mathematical equation used to solve quadratic equations, which are polynomial equations of the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are real numbers and $a \neq 0$. This formula provides a systematic way to find the solutions, or roots, of a quadratic equation.
Quotient Property: The quotient property is a fundamental concept in algebra that describes the relationship between division and exponents. It states that when a variable raised to a power is divided by the same variable raised to a lesser power, the result is the variable raised to the difference between the two exponents.
Power Property: The power property, also known as the power rule, is a fundamental concept in algebra that describes how to simplify and manipulate expressions involving exponents. It allows for the efficient calculation of powers of powers, products of powers, and quotients of powers.
Product Property: The product property is a fundamental concept in algebra that describes the relationship between the product of two expressions and the individual terms within those expressions. It is a crucial tool for simplifying and manipulating algebraic expressions, particularly when dealing with equations and inequalities.
Radical Equations: Radical equations are algebraic equations that contain at least one radical expression, such as a square root or higher-order root. These equations require special techniques to solve, as the presence of radicals introduces additional complexities compared to solving linear or polynomial equations.
Rational Exponents: Rational exponents are a way of representing fractional or negative exponents using a combination of a base number and an exponent that is a rational number, such as a fraction or a negative value. They provide a generalized way to represent and perform operations with exponents that go beyond the traditional whole number exponents.
Substitution Method: The substitution method is a technique used to solve systems of linear equations, systems of nonlinear equations, and other types of equations by substituting one variable in terms of another. This method involves isolating a variable in one equation and then substituting that expression into the other equation(s) to solve for the remaining variable(s).
Sum of Cubes: The sum of cubes is a mathematical expression that represents the sum of the cubes of two or more numbers. It is an important concept in various areas of mathematics, including factoring polynomials, solving certain types of equations, and understanding the behavior of power functions and polynomial functions.
Trinomials: A trinomial is a polynomial expression with three terms, typically in the form $ax^2 + bx + c$, where $a$, $b$, and $c$ are coefficients. Trinomials are a fundamental concept in algebra and are closely related to other types of equations, such as quadratic equations.
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Glossary
2.7 Linear Inequalities and Absolute Value Inequalities