A-REI: Reasoning with Equations and Inequalities

Reasoning with Equations and Inequalities

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Reasoning with Equations and Inequalities covers the justification and solution of equations and systems across multiple forms: linear, quadratic, and exponential. Methods include properties of equality, factoring, completing the square, the quadratic formula, and graphing. Systems of equations are solved algebraically and graphically, with matrix methods available at advanced levels.

A-REI.1Apply properties of mathematics to justify steps in solving equations in one variable.A-REI.10Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).A-REI.11Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.A-REI.12Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.A-REI.2Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.A-REI.3Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.A-REI.4Solve quadratic equations in one variable.A-REI.5Show that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.A-REI.6Solve systems of linear equations exactly and approximately, e.g., with graphs or algebraically, focusing on pairs of linear equations in two variables.A-REI.7Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = –3x and the circle x^2 + y^2 = 3.A-REI.8(+) Represent a system of linear equations as a single matrix equation in a vector variable.A-REI.9(+) Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater).
Example Problems
Solve the equation:
Graph:
What is the area of the region between the graphs of and ?
Is a solution of ?
Solve for x:
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