For example, one can add the same constant to both sides without changing the solutions, but squaring both sides might lead to extraneous solutions. Strategic competence in solving includes looking ahead for productive manipulations and anticipating the nature and number of solutions. Some equations have no solutions in a given number system, but have a solution in a larger system.
The same solution techniques used to solve equations can be used to rearrange formulas. Inequalities can be solved by reasoning about the properties of inequality. Many, but not all, of the properties of equality continue to hold for inequalities and can be useful in solving them.
Connections to Functions and Modeling. Expressions can define functions, and equivalent expressions define the same function.
Asking when two functions have the same value for the same input leads to an equation; graphing the two functions allows for finding approximate solutions of the equation. Converting a verbal description to an equation, inequality, or system of these is an essential skill in modeling.
Video Map Search Filter by: Algebra Overview Seeing Structure in Expressions Interpret the structure of expressions Write expressions in equivalent forms to solve problems Arithmetic with Polynomials and Rational Functions Perform arithmetic operations on polynomials Understand the relationship between zeros and factors of polynomials Use polynomial identities to solve problems Rewrite rational functions Creating Equations Create equations that describe numbers or relationships Reasoning with Equations and Inequalities Understand solving equations as a process of reasoning and explain the reasoning Solve equations and inequalities in one variable Solve systems of equations Represent and solve equations and inequalities graphically Mathematical Practices Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively.
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Construct viable arguments and critique the reasoning of others. Use appropriate tools strategically.
Look for and make use of structure. Functions Finding inverse functions: Functions Verifying that functions are inverses: Functions Determining whether a function is invertible: What are the imaginary numbers?
Complex numbers What are the complex numbers? Complex numbers The complex plane: Adding and subtracting complex numbers: Complex numbers Multiplying complex numbers: Adding and subtracting polynomials: Arithmetic with polynomials Multiplying polynomials: Arithmetic with polynomials Long division of polynomials: Synthetic division of polynomials: Arithmetic with polynomials Practice dividing polynomials with remainders: Arithmetic with polynomials Polynomial Remainder Theorem: Polynomials Understanding the binomial theorem: Polynomials Factoring polynomials - Quadratic forms: Polynomials Factoring polynomials - Special product forms: Polynomials Advanced polynomial factorization methods: Polynomials Proving polynomial identities: Polynomials Polynomial identities with complex numbers: Quadratic equations with complex numbers: Polynomials The Fundamental Theorem of Algebra: Polynomials Finding zeros of polynomials: Polynomials Zeros of polynomials and their graphs: Polynomials End behavior of polynomial functions: Polynomials Graphs of polynomials: Polynomials Introduction to symmetry of functions: Polynomials Symmetry of polynomial functions: Radical relationships Extraneous solutions of radical equations: Radical relationships Solving cube-root equations: Domain of radical functions: Radical relationships Graphs of radical functions: Rational relationships Nested fractions: Rational relationships Solving rational equations: Direct and inverse variation: Rational relationships End behavior of rational functions: Rational relationships Discontinuities of rational functions:
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