9.6 Systems of Inequalities
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This document provides an overview of systems of inequalities and how to graph them. It discusses representing inequalities in slope-intercept form and plotting the solution regions based on whether the line is above or below the x-axis. Examples are provided of graphing individual inequalities and systems of inequalities. The solution to a system is the overlapping region where both inequalities are satisfied. Students are assigned practice problems graphing and solving systems of inequalities.
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9.5 Nonlinear Systems of Equations
This document discusses methods for solving nonlinear systems of equations, including elimination, substitution, and graphing. It provides examples of solving systems by each method and writing the solutions as ordered pairs. Nonlinear systems contain at least one equation that is not linear. Elimination transforms equations so variables can be eliminated. Substitution solves one equation for a variable and substitutes into the other equations. Graphing can also show the intersection points that are the solutions. The document emphasizes being able to use any solving method and explains solutions may be irrational numbers. It concludes with assigning practice problems from the textbook.
9.3 Determinant Solution of Linear Systems
Calculate the determinant of an n x n matrix
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This document discusses solving nonlinear systems of equations in three ways: elimination, substitution, and graphing. It provides examples of solving systems by each method and discusses key steps like eliminating variables, substituting values, and finding the intersection point(s) of graphs. Absolute value equations are also addressed. Students are expected to be able to solve nonlinear systems by any method, as questions may require non-graphical solutions for irrational answers. A classwork and quiz assignment is provided to practice these skills.
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The document provides examples for solving systems of linear equations by substitution. It explains the substitution method and walks through three examples step-by-step. The key steps are: 1) solve one equation for one variable, 2) substitute the expression into the other equation, 3) solve the resulting equation for the second variable, 4) substitute values back into the original equations to check. An example application compares the total costs of two cell phone plans over time.
1.4 Quadratic Equations
This document summarizes key concepts about quadratic equations, including:
- Quadratic equations can be solved by factoring, completing the square, or using the quadratic formula.
- Completing the square involves manipulating the equation into a perfect square trinomial form.
- The quadratic formula provides the solutions to any quadratic equation in standard form.
- Cubic equations that are the sum or difference of cubes can be factored and solved.
- Literal quadratic equations can be solved for a specified variable using techniques like the square root property or quadratic formula.
- The discriminant determines whether the solutions to a quadratic equation are rational, irrational, or complex numbers.
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This document provides instructions for solving systems of equations using elimination. It demonstrates eliminating variables by adding or subtracting equations. Sample systems are worked through, showing the steps of identifying which variable to eliminate, combining the equations accordingly, solving for one variable, then substituting back into the original equations to solve for the other. The solutions are checked in both equations to verify they satisfy the system.
February 13, 2015
1. The document outlines the day's math lesson which includes reviewing systems of equations solutions, solving 3x3 systems, and completing yesterday's class work.
2. It provides examples and steps for solving systems of equations by graphing, elimination, and substitution. Equations are presented in standard form and slope-intercept form.
3. Solving 3x3 systems is discussed, noting they cannot be graphed since they exist in three dimensions. The substitution method is demonstrated through an example.
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This document provides examples of solving systems of linear equations by elimination. It explains the steps:
1) Write the system so that like terms are aligned.
2) Eliminate one variable by adding or subtracting the equations.
3) Substitute the value into one equation to solve for the other variable.
4) Write the solution as an ordered pair and check.
It shows how to multiply equations by a number to produce opposite coefficients for elimination. Examples demonstrate solving systems by addition, subtraction, and multiplication.
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This document summarizes key concepts about quadratic equations, including:
- Quadratic equations can be solved by factoring, completing the square, or using the quadratic formula.
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1. The document outlines the day's math lesson which includes reviewing systems of equations solutions, solving 3x3 systems, and completing yesterday's class work.
2. It provides examples and steps for solving systems of equations by graphing, elimination, and substitution. Equations are presented in standard form and slope-intercept form.
3. Solving 3x3 systems is discussed, noting they cannot be graphed since they exist in three dimensions. The substitution method is demonstrated through an example.
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This document provides examples of solving systems of linear equations by elimination. It explains the steps:
1) Write the system so that like terms are aligned.
2) Eliminate one variable by adding or subtracting the equations.
3) Substitute the value into one equation to solve for the other variable.
4) Write the solution as an ordered pair and check.
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The document discusses methods for solving simultaneous linear equations, including elimination and substitution.
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6 2 Solving Systems with substitution
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1) Put the equations in standard form if needed
2) Determine that y can be eliminated by multiplying the top equation by 3
3) Multiply the top equation by 3 and add it to the bottom equation to eliminate y
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This document contains information about a group project from the Universidad Politécnica Territorial del Estado Lara Andrés Eloy Blanco in Barquisimeto, Venezuela. The document lists the group members, Keishmer Amaro and Heycker Cuicas, and indicates they are studying hygiene and occupational safety in section 0102.
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The test this week will cover content from Assignments 0.1 through 0.7. Students are responsible for all content, whether or not they were present for that class. The test questions will be similar to sample questions provided. Review solutions are posted online but will be removed the morning of each test.
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This document contains information about a group project from the Universidad Politécnica Territorial del Estado Lara Andrés Eloy Blanco in Barquisimeto, Venezuela. The document lists the group members, Keishmer Amaro and Heycker Cuicas, and indicates they are studying hygiene and occupational safety in section 0102.
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1.5 Quadratic Equations.pdf
This document provides instruction on factoring polynomials and quadratic equations. It begins by reviewing factoring techniques like finding the greatest common factor and factoring trinomials and binomials. Examples are provided to demonstrate the factoring methods. The document then discusses solving quadratic equations by factoring, putting the equation in standard form, and setting each factor equal to zero. An example problem demonstrates solving a quadratic equation through factoring. The document concludes by assigning homework and an optional reading for the next class.
3.2 Graphs of Functions
* Connect functions to their graphs
* Graph piecewise-defined functions
* Graph absolute value functions
* Graph greatest-integer functions
* Interpret graphs
* Use the vertical line test to determine a function
3.2 Graphs of Functions
* Connect functions to their graphs
* Graph piecewise-defined functions
* Graph absolute value functions
* Graph greatest-integer functions
* Interpret graphs
* Use the vertical line test to determine a function
3.1 Functions
* Introduce functions and function notation
* Develop skills in constructing and interpreting the graphs of functions
* Learn to apply this knowledge in a variety of situations
2.5 Transformations of Functions
* Recognize graphs of common functions.
* Graph functions using vertical and horizontal shifts.
* Graph functions using reflections about the x-axis and the y-axis.
* Graph functions using compressions and stretches.
* Combine transformations.
1.5 Quadratic Equations (Review)
This document provides instruction on factoring quadratic equations. It begins by reviewing factoring polynomials and trinomials. It then discusses factoring binomials using difference of squares, sum/difference of cubes, and other patterns. Finally, it explains that a quadratic equation can be solved by factoring if it can be written as a product of two linear factors. An example demonstrates factoring a quadratic equation by finding the two values that make each factor equal to zero.
2.1 Basics of Functions and Their Graphs
This document provides an overview of functions and their graphs. It defines what constitutes a function, discusses domain and range, and how to identify functions using the vertical line test. Key points covered include:
- A function is a relation where each input has a single, unique output
- The domain is the set of inputs and the range is the set of outputs
- Functions can be represented by ordered pairs, graphs, or equations
- The vertical line test identifies functions as those where a vertical line intersects the graph at most once
- Intercepts occur where the graph crosses the x or y-axis
9.6 Binomial Theorem
The document discusses the binomial theorem, which provides a formula for expanding binomial expressions of the form (a + b)^n. It gives the formula for finding the coefficient of the term containing b^r as nCr. Several examples are worked out applying the binomial theorem to expand binomial expressions and find specific terms. Factorial notation is introduced for writing the coefficients. The document also discusses using calculators and Desmos to evaluate binomial coefficients. Practice problems are assigned from previous sections.
13.3 Venn Diagrams & Two-Way Tables
The document discusses using Venn diagrams and two-way tables to organize data and calculate probabilities. It provides examples of completing Venn diagrams and two-way tables based on survey data about students' activities. It then uses the tables and diagrams to calculate probabilities of different outcomes. The examples illustrate how to set up and use these visual representations of categorical data.
13.2 Independent & Dependent Events
* Identify events as independent or dependent
* Calculate the probabilities of independent and dependent events
9.5 Counting Principles
* Solve counting problems using the Addition Principle.
* Solve counting problems using the Multiplication Principle.
* Solve counting problems using permutations involving n distinct objects.
* Solve counting problems using combinations.
* Find the number of subsets of a given set.
* Solve counting problems using permutations involving n non-distinct objects.
13.1 Geometric Probability
* Calculate geometric probabilities
* Use geometric probability to predict results in real world situations.
9.4 Series and Their Notations
* Use summation notation.
* Use the formula for the sum of the first n terms of an arithmetic series.
* Use the formula for the sum of the first n terms of a geometric series.
* Use the formula for the sum of an infinite geometric series.
* Solve annuity problems.
9.3 Geometric Sequences
* Find the common ratio for a geometric sequence.
* List the terms of a geometric sequence.
* Use a recursive formula for a geometric sequence.
* Use an explicit formula for a geometric sequence.
9.2 Arithmetic Sequences
* Find the common difference for an arithmetic sequence.
* Write terms of an arithmetic sequence.
* Use a recursive formula for an arithmetic sequence.
* Use an explicit formula for an arithmetic sequence.
9.1 Sequences and Their Notations
* Write the terms of a sequence defined by an explicit formula.
* Write the terms of a sequence defined by a recursive formula.
* Use factorial notation.
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9.6 Systems of Inequalities
- 1. 9.6 Systems of Inequalities Chapter 9 Systems and Matrices
- 2. Concepts and Objectives ⚫ Systems of Inequalities ⚫ Graph systems of inequalities ⚫ Identify solutions to systems of inequalities
- 3. Graphing Review ⚫ To graph a linear inequality, put the inequality into slope-intercept form. ⚫ Plot the y-intercept and count the slope from there (rise over run) Symbol Line Shade < below > above below above
- 4. Graphing Review ⚫ Example: Graph the solution to + 2 3 6x y
- 5. Graphing Review ⚫ Example: Graph the solution to + 2 3 6x y − +3 2 6y x − + 2 2 3 y x
- 6. Graphing Review ⚫ Example: What is the solution to ? −2 2y x
- 7. Graphing Review ⚫ Example: What is the solution to ? Vertex: (0, –2) −2 2y x
- 8. Systems of Inequalities ⚫ The solution to a system of inequalities will be the graph of the overlap between the two (or more) inequalities. ⚫ You must graph the system to show the solution. ⚫ If the inequalities do not overlap, then there is no solution to the system.
- 9. Systems of Inequalities ⚫ Example: What is the solution to the system? − − + 2 4 3 2 x y x y
- 10. Systems of Inequalities ⚫ Example: What is the solution to the system? Inequality #1: − − + 2 4 3 2 x y x y − − −2 4y x + 1 2 2 y x
- 11. Systems of Inequalities ⚫ Example: What is the solution to the system? Inequality #2: − − + 2 4 3 2 x y x y − +3 2y x
- 12. Systems of Inequalities ⚫ Example: What is the solution to the system? − + 2 1 3 2 4 x y x y
- 13. Systems of Inequalities ⚫ Example: What is the solution to the system? Inequality #1 − + 2 1 3 2 4 x y x y − − +2 1y x −2 1y x
- 14. Systems of Inequalities ⚫ Example: What is the solution to the system? Inequality #2 − + 2 1 3 2 4 x y x y − +2 3 4y x − + 3 2 2 y x
- 15. Classwork ⚫ College Algebra (9.6 Assignment) ⚫ Page 904: 12-16 (even), page 876: 36-46 (even), page 850: 48-52 (even)