Interval notation
•Download as PPT, PDF•
0 likes•213 views
Interval notation is used to represent sets of numbers graphically or through inequalities. The objectives are to become familiar with interval notation and to use it to represent sets of numbers given in graphical or inequality form in 3 sentences or fewer.
Report
Share
Report
Share
Recommended
41 interval notation and review on inequalities
The document discusses inequalities and comparisons of numbers on the real number line. It explains that numbers to the right are greater than numbers to the left. Various inequality symbols like <, >, ≤ are introduced to represent relationships between numbers. Intervals on the real number line, such as -3 ≤ x < 2, are defined. Basic algebra rules for retaining inequality signs under addition, subtraction, multiplication, and division are also covered.
El muchacho de los libros
Kyle llevaba todos sus libros a casa un viernes con la intención de suicidarse ese fin de semana. Fue atacado por otros estudiantes, pero el narrador lo ayudó y se hicieron amigos. Años después, en el discurso de graduación, Kyle reveló su intención de suicidio y cómo su amistad con el narrador lo salvó.
Capítulo 11.2
Este resumen proporciona una sinopsis del capítulo en 3 oraciones o menos:
Luce asume el cuerpo de Lys, una princesa en el siglo XVIII, y es seleccionada por el joven y enfermizo Rey Louis XV para ser su posible esposa. Sin embargo, Luce ama a Daniel, quien es un sirviente en esta vida. Luce escapa de la recepción con el rey y encuentra a Daniel en el balcón, donde se reúnen después de tanto tiempo separados.
Finalprojectnlu
The art pieces chosen from the Art Institute of Chicago depict scenes and feelings from 19th century Europe that appeal to the presenter. Five items will be displayed thematically in a gallery space designed to resemble a 19th century room: a bust will greet visitors near the entrance while two photographs flank a couch, and two additional pieces are placed near the exit, one on each side of the door. Though varying in medium, all pieces have warm tones and expressiveness that evoke 19th century European lifestyle and romance in a way that is suitable for modern decorative display.
Psychology test
This document presents a 4 question personality test about preferences for animals, descriptions of words, relating colors to important people, and choosing a favorite number and day of the week. It claims the answers will reveal priorities, personalities, relationships and predict how many people you share it with will impact your life. The document encourages honesty and finishing the test before viewing the interpretations.
Ayeh powerpoint....
This document discusses management functions and organizing a sports meet. It outlines the six main functions of management: planning, organizing, staffing, leading, motivating, and controlling. It then lists the key committees needed to organize a sports meet: finance and resources, program and invitation, peace and order, physical facilities, equipment, prizes and awards, and technical committee. Each committee is described briefly in one sentence. The document concludes by emphasizing the importance of putting God first in all undertakings.
Brand new world
Social media has over 700 million users worldwide that are increasing at a devastating rate, allowing for easy and meaningful connections through platforms that can work wonders when leveraged in the best way.
Interval Notation
Interval notation is a way to describe sets of real numbers using brackets and parentheses. It is an alternative to set notation. Round parentheses correspond to open intervals that exclude endpoint values, while square brackets correspond to closed intervals that include endpoints. Common interval notation examples include (a,b) for open intervals and [a,b] for closed intervals. This notation can be used to clearly define the domain and range of functions and relations.
Recommended
41 interval notation and review on inequalities
The document discusses inequalities and comparisons of numbers on the real number line. It explains that numbers to the right are greater than numbers to the left. Various inequality symbols like <, >, ≤ are introduced to represent relationships between numbers. Intervals on the real number line, such as -3 ≤ x < 2, are defined. Basic algebra rules for retaining inequality signs under addition, subtraction, multiplication, and division are also covered.
El muchacho de los libros
Kyle llevaba todos sus libros a casa un viernes con la intención de suicidarse ese fin de semana. Fue atacado por otros estudiantes, pero el narrador lo ayudó y se hicieron amigos. Años después, en el discurso de graduación, Kyle reveló su intención de suicidio y cómo su amistad con el narrador lo salvó.
Capítulo 11.2
Este resumen proporciona una sinopsis del capítulo en 3 oraciones o menos:
Luce asume el cuerpo de Lys, una princesa en el siglo XVIII, y es seleccionada por el joven y enfermizo Rey Louis XV para ser su posible esposa. Sin embargo, Luce ama a Daniel, quien es un sirviente en esta vida. Luce escapa de la recepción con el rey y encuentra a Daniel en el balcón, donde se reúnen después de tanto tiempo separados.
Finalprojectnlu
The art pieces chosen from the Art Institute of Chicago depict scenes and feelings from 19th century Europe that appeal to the presenter. Five items will be displayed thematically in a gallery space designed to resemble a 19th century room: a bust will greet visitors near the entrance while two photographs flank a couch, and two additional pieces are placed near the exit, one on each side of the door. Though varying in medium, all pieces have warm tones and expressiveness that evoke 19th century European lifestyle and romance in a way that is suitable for modern decorative display.
Psychology test
This document presents a 4 question personality test about preferences for animals, descriptions of words, relating colors to important people, and choosing a favorite number and day of the week. It claims the answers will reveal priorities, personalities, relationships and predict how many people you share it with will impact your life. The document encourages honesty and finishing the test before viewing the interpretations.
Ayeh powerpoint....
This document discusses management functions and organizing a sports meet. It outlines the six main functions of management: planning, organizing, staffing, leading, motivating, and controlling. It then lists the key committees needed to organize a sports meet: finance and resources, program and invitation, peace and order, physical facilities, equipment, prizes and awards, and technical committee. Each committee is described briefly in one sentence. The document concludes by emphasizing the importance of putting God first in all undertakings.
Brand new world
Social media has over 700 million users worldwide that are increasing at a devastating rate, allowing for easy and meaningful connections through platforms that can work wonders when leveraged in the best way.
Interval Notation
Interval notation is a way to describe sets of real numbers using brackets and parentheses. It is an alternative to set notation. Round parentheses correspond to open intervals that exclude endpoint values, while square brackets correspond to closed intervals that include endpoints. Common interval notation examples include (a,b) for open intervals and [a,b] for closed intervals. This notation can be used to clearly define the domain and range of functions and relations.
3.4 linear programming
The document discusses linear programming problems. It provides two examples of finding the minimum and maximum values of objective functions subject to certain constraints. In the first example, the objective is to minimize or maximize the function f(x,y)=3x-2y within the constraints 1≤x≤5, y≥2, and y≤x+3. In the second example, the objective is to minimize or maximize the function f(x,y)=4x+3y within the constraints y≥-x+2, y≤(1/4)x+2, and y≥2x-5. Both examples solve the problems by finding the vertices of the constrained regions and calculating the objective
3.3 a writing systems of linear inequalities
The document discusses how to graph and write systems of linear inequalities. It explains how to graph systems by determining whether lines should be dotted or solid based on the inequality symbols, and whether shading should be above or below lines based on 'y >' or 'y <'. It also explains how to write the system of inequalities given a graph by writing equations for each line and changing the equations to inequalities based on whether lines are dotted or solid and whether shading is above or below lines. Examples of these processes are provided.
3.3 a writing a systems of inequalities
This document discusses how to graph systems of linear inequalities by:
1. Writing each inequality in the system in slope-intercept form (y=mx+b)
2. Graphing the lines for each inequality, making dashed lines for > or < and solid for ≥ or ≤
3. Shading the correct region for each line (above for > and ≥, below for < and ≤)
4. The solution to the system is the overlapping region shaded by both inequalities.
3.3 a solving systems of inequalities
1. Systems of linear inequalities can be solved by graphing the inequalities on a coordinate plane.
2. Each inequality is written in slope-intercept form (y=mx+b) and graphed as either a solid or dotted line.
3. The region representing the solution of the system is found by shading the area of overlap between the graphs based on whether each inequality represents "above or below" the line.
3.3 solving systems of inequalities
The document discusses solving systems of linear inequalities by graphing the inequalities on a coordinate plane. It explains that linear inequalities should be written in slope-intercept form and the corresponding lines should be drawn as dotted or solid based on the inequality symbols. It also notes that the solution region includes points not on the lines and instructs the reader to shade above or below lines depending on the inequality. An example problem demonstrates these steps to find the region satisfying two inequalities.
3.2 a solving systems algebraically
Here are the steps to solve this system of linear equations using the elimination method:
1) Line up the equations:
-4x - 8y = 16
6x + 12y = -24
2) Add a multiple of one equation to the other to eliminate x:
-4x - 8y = 16
6x + 12y = -24
Add -1.5 times the top equation to the bottom:
-4x - 8y = 16
0 - 3y = -40
3) Solve for y:
-3y = -40
y = 13
4) Substitute y = 13 back into either original equation to find x:
-4x
3.2 solving systems algebraically
Here are the steps to solve this system of equations using elimination:
1) Line up the equations:
-4x - 8y = 16
6x + 12y = -24
2) Multiply the top equation by -3:
12x + 24y = -48
6x + 12y = -24
3) Add the equations to eliminate y:
18x = -72
4) Divide both sides by 18:
x = -4
5) Substitute x=-4 into either original equation to find y:
-4(-4) - 8y = 16
16 - 8y = 16
-8y = 0
y = 0
So
3.1 b solving systems graphically
This document provides instructions for students to solve systems of linear equations by graphing. Students are asked to write equations in slope-intercept form when possible, use slope and y-intercept to determine the number of solutions (one, many, or no solutions), and name each system based on its number of solutions. The document assigns textbook problems #35-52 for practice and is dated for Friday, October 7th.
3.1 a solving systems graphically
This document discusses how to graph and solve systems of linear equations in two variables by determining whether the system has one solution, many solutions, or no solution. It explains that a consistent system has a solution, an inconsistent system has no solution, an independent system has one solution, and a dependent system has infinitely many solutions. It also notes that a system will have one solution if the lines cross once, no solution if the lines are parallel, and infinitely many solutions if the lines are the same.
3.1 solving systems graphically
The document discusses solving systems of linear equations by graphing. It explains that a system of linear equations contains two or more linear equations using the same variables. To solve a linear system graphically, the two equations must be put into slope-intercept form and graphed on the same coordinate plane. The point where the lines intersect satisfies both equations and is the solution to the system. The solution must be verified by substituting the coordinates into the original equations.
3.4 a linear programming
The document discusses using linear programming to find the maximum and minimum values of an objective function given certain constraints. It provides an example of finding the minimum and maximum values of the function f(x,y)=3x-2y when constrained by y≥2, 1≤x≤5, and y≤x+3. The values are found by graphing the inequalities to determine the vertices and substituting the vertices into the objective function.
2.8 b absolute value functions
The document provides instructions for graphing and writing equations for absolute value functions. It discusses key aspects like the vertex, line of symmetry, slope, and general form of the equation. Examples are provided to demonstrate how to graph an absolute value function given its equation, and how to write the equation of an absolute value function given its graph by finding the vertex and another point.
2.8 a absolute value functions
1. The document discusses absolute value functions and how to graph and write equations for them.
2. It explains that the graph of an absolute value function is V-shaped and defines the vertex and line of symmetry.
3. Examples are provided to demonstrate how to graph an absolute value function by finding the vertex and slope and plotting points, and how to write the equation of an absolute value function given its graph.
2.8 absolute value functions
1. The document discusses absolute value functions of the form y = a|x - h| + k, where a ≠ 0.
2. It explains how to find the vertex (h, k) and line of symmetry x = h of an absolute value function graph.
3. It also describes how the graph opens up or down depending on whether a is positive or negative, and how the width compares to y = |x| depending on the value of |a|.
2.6 graphing linear inequalities
This document discusses graphing linear inequalities. It has two objectives: 1) determining if an ordered pair is a solution to an inequality, and 2) graphing linear inequalities in one and two variables.
2.5 a correlation & best fitting lines
The document discusses finding the line of best fit for a data set using linear regression. It explains that the line of best fit should have approximately the same number of points above and below the line. It also describes how the slope of the line indicates whether the correlation is positive, negative, or relatively no correlation. Finally, it introduces the correlation coefficient r as a measure of the strength of correlation, with values closer to 1 or -1 indicating a stronger correlation.
2.5 correlation & best fitting lines
This document discusses identifying correlation in data by plotting it on a scatter plot and fitting a best-fit line. It explains that a positive slope indicates positive correlation, a negative slope indicates negative correlation, and no consistent slope indicates little correlation. It also introduces the correlation coefficient r as a measure of correlation strength, with values closer to 1 or -1 indicating stronger correlation.
2.4 writing linear equations
This document discusses writing and working with linear equations in slope-intercept form (y=mx+b). It provides examples of:
1) Writing linear equations from the slope (m) and y-intercept (b)
2) Finding the slope and y-intercept of given linear equations
3) Writing linear equations given two points or given a slope and point.
2.4 writing equations of lines
1) This document discusses writing linear equations in slope-intercept form (y=mx+b) given different information: the slope (m) and y-intercept (b), a graph, the slope and a point, or two points.
2) It provides examples of writing equations of lines given: the slope and y-intercept, a graph, the slope and a point, two points, or the x- and y-intercepts.
3) The key steps are to identify the slope (m) and y-intercept (b) from the information provided, then substitute those values into the slope-intercept form equation to write the linear equation.
2.3 linear equations
The document discusses linear equations and how to graph them. It defines linear equations as having variables with exponents of 1 that are added or subtracted. It explains how to identify the slope and y-intercept of a linear equation in slope-intercept form (y=mx+b) in order to graph it as a line on the coordinate plane. Key steps include solving for y and identifying the slope (m) and y-intercept (b). Examples are provided to demonstrate finding intercepts and graphing various linear equations.
More Related Content
More from fthrower
3.4 linear programming
The document discusses linear programming problems. It provides two examples of finding the minimum and maximum values of objective functions subject to certain constraints. In the first example, the objective is to minimize or maximize the function f(x,y)=3x-2y within the constraints 1≤x≤5, y≥2, and y≤x+3. In the second example, the objective is to minimize or maximize the function f(x,y)=4x+3y within the constraints y≥-x+2, y≤(1/4)x+2, and y≥2x-5. Both examples solve the problems by finding the vertices of the constrained regions and calculating the objective
3.3 a writing systems of linear inequalities
The document discusses how to graph and write systems of linear inequalities. It explains how to graph systems by determining whether lines should be dotted or solid based on the inequality symbols, and whether shading should be above or below lines based on 'y >' or 'y <'. It also explains how to write the system of inequalities given a graph by writing equations for each line and changing the equations to inequalities based on whether lines are dotted or solid and whether shading is above or below lines. Examples of these processes are provided.
3.3 a writing a systems of inequalities
This document discusses how to graph systems of linear inequalities by:
1. Writing each inequality in the system in slope-intercept form (y=mx+b)
2. Graphing the lines for each inequality, making dashed lines for > or < and solid for ≥ or ≤
3. Shading the correct region for each line (above for > and ≥, below for < and ≤)
4. The solution to the system is the overlapping region shaded by both inequalities.
3.3 a solving systems of inequalities
1. Systems of linear inequalities can be solved by graphing the inequalities on a coordinate plane.
2. Each inequality is written in slope-intercept form (y=mx+b) and graphed as either a solid or dotted line.
3. The region representing the solution of the system is found by shading the area of overlap between the graphs based on whether each inequality represents "above or below" the line.
3.3 solving systems of inequalities
The document discusses solving systems of linear inequalities by graphing the inequalities on a coordinate plane. It explains that linear inequalities should be written in slope-intercept form and the corresponding lines should be drawn as dotted or solid based on the inequality symbols. It also notes that the solution region includes points not on the lines and instructs the reader to shade above or below lines depending on the inequality. An example problem demonstrates these steps to find the region satisfying two inequalities.
3.2 a solving systems algebraically
Here are the steps to solve this system of linear equations using the elimination method:
1) Line up the equations:
-4x - 8y = 16
6x + 12y = -24
2) Add a multiple of one equation to the other to eliminate x:
-4x - 8y = 16
6x + 12y = -24
Add -1.5 times the top equation to the bottom:
-4x - 8y = 16
0 - 3y = -40
3) Solve for y:
-3y = -40
y = 13
4) Substitute y = 13 back into either original equation to find x:
-4x
3.2 solving systems algebraically
Here are the steps to solve this system of equations using elimination:
1) Line up the equations:
-4x - 8y = 16
6x + 12y = -24
2) Multiply the top equation by -3:
12x + 24y = -48
6x + 12y = -24
3) Add the equations to eliminate y:
18x = -72
4) Divide both sides by 18:
x = -4
5) Substitute x=-4 into either original equation to find y:
-4(-4) - 8y = 16
16 - 8y = 16
-8y = 0
y = 0
So
3.1 b solving systems graphically
This document provides instructions for students to solve systems of linear equations by graphing. Students are asked to write equations in slope-intercept form when possible, use slope and y-intercept to determine the number of solutions (one, many, or no solutions), and name each system based on its number of solutions. The document assigns textbook problems #35-52 for practice and is dated for Friday, October 7th.
3.1 a solving systems graphically
This document discusses how to graph and solve systems of linear equations in two variables by determining whether the system has one solution, many solutions, or no solution. It explains that a consistent system has a solution, an inconsistent system has no solution, an independent system has one solution, and a dependent system has infinitely many solutions. It also notes that a system will have one solution if the lines cross once, no solution if the lines are parallel, and infinitely many solutions if the lines are the same.
3.1 solving systems graphically
The document discusses solving systems of linear equations by graphing. It explains that a system of linear equations contains two or more linear equations using the same variables. To solve a linear system graphically, the two equations must be put into slope-intercept form and graphed on the same coordinate plane. The point where the lines intersect satisfies both equations and is the solution to the system. The solution must be verified by substituting the coordinates into the original equations.
3.4 a linear programming
The document discusses using linear programming to find the maximum and minimum values of an objective function given certain constraints. It provides an example of finding the minimum and maximum values of the function f(x,y)=3x-2y when constrained by y≥2, 1≤x≤5, and y≤x+3. The values are found by graphing the inequalities to determine the vertices and substituting the vertices into the objective function.
2.8 b absolute value functions
The document provides instructions for graphing and writing equations for absolute value functions. It discusses key aspects like the vertex, line of symmetry, slope, and general form of the equation. Examples are provided to demonstrate how to graph an absolute value function given its equation, and how to write the equation of an absolute value function given its graph by finding the vertex and another point.
2.8 a absolute value functions
1. The document discusses absolute value functions and how to graph and write equations for them.
2. It explains that the graph of an absolute value function is V-shaped and defines the vertex and line of symmetry.
3. Examples are provided to demonstrate how to graph an absolute value function by finding the vertex and slope and plotting points, and how to write the equation of an absolute value function given its graph.
2.8 absolute value functions
1. The document discusses absolute value functions of the form y = a|x - h| + k, where a ≠ 0.
2. It explains how to find the vertex (h, k) and line of symmetry x = h of an absolute value function graph.
3. It also describes how the graph opens up or down depending on whether a is positive or negative, and how the width compares to y = |x| depending on the value of |a|.
2.6 graphing linear inequalities
This document discusses graphing linear inequalities. It has two objectives: 1) determining if an ordered pair is a solution to an inequality, and 2) graphing linear inequalities in one and two variables.
2.5 a correlation & best fitting lines
The document discusses finding the line of best fit for a data set using linear regression. It explains that the line of best fit should have approximately the same number of points above and below the line. It also describes how the slope of the line indicates whether the correlation is positive, negative, or relatively no correlation. Finally, it introduces the correlation coefficient r as a measure of the strength of correlation, with values closer to 1 or -1 indicating a stronger correlation.
2.5 correlation & best fitting lines
This document discusses identifying correlation in data by plotting it on a scatter plot and fitting a best-fit line. It explains that a positive slope indicates positive correlation, a negative slope indicates negative correlation, and no consistent slope indicates little correlation. It also introduces the correlation coefficient r as a measure of correlation strength, with values closer to 1 or -1 indicating stronger correlation.
2.4 writing linear equations
This document discusses writing and working with linear equations in slope-intercept form (y=mx+b). It provides examples of:
1) Writing linear equations from the slope (m) and y-intercept (b)
2) Finding the slope and y-intercept of given linear equations
3) Writing linear equations given two points or given a slope and point.
2.4 writing equations of lines
1) This document discusses writing linear equations in slope-intercept form (y=mx+b) given different information: the slope (m) and y-intercept (b), a graph, the slope and a point, or two points.
2) It provides examples of writing equations of lines given: the slope and y-intercept, a graph, the slope and a point, two points, or the x- and y-intercepts.
3) The key steps are to identify the slope (m) and y-intercept (b) from the information provided, then substitute those values into the slope-intercept form equation to write the linear equation.
2.3 linear equations
The document discusses linear equations and how to graph them. It defines linear equations as having variables with exponents of 1 that are added or subtracted. It explains how to identify the slope and y-intercept of a linear equation in slope-intercept form (y=mx+b) in order to graph it as a line on the coordinate plane. Key steps include solving for y and identifying the slope (m) and y-intercept (b). Examples are provided to demonstrate finding intercepts and graphing various linear equations.
More from fthrower (20)
Interval notation
- 1. INTERVAL NOTATION NO CALCULATOR OR CLICKER! Today’s objectives: 1. I will become familiar with interval notation. 2. I will be able to use interval notation to represent sets of numbers given graphically or in inequality form.