Inequalities
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This document discusses inequalities and provides examples of solving various inequality expressions. It begins by giving an example of a negative number that is not closed under multiplication. It then defines common inequality symbols like less than, greater than, less than or equal to, and greater than or equal to. The document proceeds to give examples of solving linear inequalities assuming the domain is all real numbers, including compound inequalities involving absolute value expressions. It concludes with examples for an exit slip problem set involving solving several inequalities.
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Inequalities
This document provides information and examples about solving inequalities:
1) It gives examples of solving linear equations with one variable and two variables.
2) It explains the symbols used for different inequality types and how to graph them, including shading the correct side of the inequality.
3) It provides rules for solving inequalities, such as reversing the inequality sign when multiplying or dividing both sides by a negative number.
4) Multiple examples are worked through step-by-step of solving and graphing various inequalities with one variable and multiple variables.
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For more instructional resources, CLICK me here! 👇👇👇
https://tinyurl.com/y9muob6q
LIKE and FOLLOW me here! 👍👍👍
https://tinyurl.com/ycjp8r7u
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To factor trinomials of the form ax^2 + bx + c, look at the signs of the coefficients and find two factors whose product equals c. The factors must be the same sign as b or opposite signs. Examples provided factor 6x^2 + 7x - 24 as (6x + 6)(x - 4) and 4x^2 - 12x - 7 as (4x - 7)(x + 1). Practice problems are assigned.
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This document provides information and examples about solving inequalities:
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3) It provides rules for solving inequalities, such as reversing the inequality sign when multiplying or dividing both sides by a negative number.
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This document discusses factoring polynomials that are the difference of two squares using the formula a2 - b2 = (a + b)(a - b). It provides examples of factoring polynomials like x2 - 16, 9x2 - 100, and 36m2 - 49n4. It also lists 10 practice problems for factoring polynomials that are differences of two squares and references where readers can learn more.
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3) The sum or difference of two cubes can be factored into a binomial times a trinomial using specific formulas.
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For more instructional resources, CLICK me here!
https://tinyurl.com/y9muob6q
LIKE and FOLLOW me here!
https://tinyurl.com/ycjp8r7u
https://tinyurl.com/ybo27k2u
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Introduction to coordinate geometry
Any point on a plane can be named with two numbers called coordinates - an x-coordinate for the distance from the origin along the x-axis and a y-coordinate for the distance along the y-axis. The coordinate grid is divided into four quadrants by the x and y axes, and points can be plotted in the grid and identified by their coordinates and quadrant. Linear equations can be graphed by making a table of solutions and plotting the points, then drawing a straight line through them.
F(x) terminology
This document contains terminology and examples related to functions. It defines f(x) as "the value of y in function f, where the independent variable is x". It provides examples of evaluating functions like f(x) = -7x + 100 for different x values. It also gives examples of composite functions like f(f(x)) and using different functions like f(x) and g(x) together. The examples relate to word problems about distances over time on a boat trip.
Translating words into symbols
This document provides examples of translating word problems into mathematical equations and then solving those equations. It contains sample word problems that translate to equations involving addition, subtraction, multiplication and division. It also includes guidance on common phrases like "less than" and their symbolic representations. Finally, it provides two word problems as examples for students to practice translating words into mathematical symbols.
Solving equations with variables on both sides & translating words into symbols
This document contains sample math problems and notes about solving equations with variables on both sides. It includes examples of solving equations like 5y - 3y + 13 = 26, 3s + 2 = 8, 2d + 3d = 15, and 2(w -1) + 8 = 16. The notes provide instructions for solving equations where variables are on both sides, such as 6x = 4x + 18, 3y = 15 - 2y, and 7(a - 2) - 6 = 2a + 8 + a. The document also references a Solving Equations BINGO activity and exit slip homework.
4.3 cramer’s rule
The document provides information about solving systems of linear equations using Cramer's rule, including:
- Cramer's rule uses determinants of the coefficient matrix to solve systems of linear equations.
- For a 2x2 system, if the determinant of the coefficient matrix is not equal to 0, there is exactly one solution.
- The solutions are found by taking the determinants of the numerator matrices (with one column replaced by the constants) divided by the determinant of the coefficient matrix.
- Several examples demonstrate solving 2x2 systems using Cramer's rule.
Solving percent equations
The document discusses percentages and solving percentage equations. It provides examples of how to write percentage equations for word problems, such as finding what percentage one number is of another or finding the total given a percentage. It demonstrates solving equations to find percentages and totals in various contexts like test scores, class sizes, sales, and more. Percentages allow for easier comparison of quantities and are used in many everyday situations like grades, weather, nutrition labels, and shopping.
Decimals, fractions, and percents
This document discusses how to convert between decimals, fractions, and percents. It provides examples for converting non-repeating and repeating decimals to fractions by dropping the decimal point and putting the number over the place value. It also demonstrates how to convert fractions to decimals by dividing the numerator by the denominator and moving the decimal point. Additionally, it shows how to convert decimals to percents by moving the decimal point two places to the right and adding the percent sign, and how to convert percents to decimals by moving the decimal point two places to the left and dropping the percent sign.
The matrix
The document contains information about the reciprocal of 1/9, the value of |-8+-2|, the difference between a January wind chill of -8 degrees and an August heat index of 108 degrees. It also contains a summary of the 1999 film The Matrix and definitions of a matrix and rules for adding, subtracting and multiplying matrices by scalars.
Simplifying expressions, solving, domain and range
This document appears to be notes from a math class covering simplifying expressions, solving equations, and domain and range. It includes examples of irrational and imaginary numbers, naming polynomials by degree and terms, multiplying binomials, homework assignments on the topics, and notes on using the vertical line test to determine if a relation is a function by checking if each x-value has exactly one y-value. Sketching graphs of functions given domains and ranges is also mentioned.
04.07.10 more volume exercises
The document provides instructions to sketch and calculate the volumes of different shapes, including a rectangular prism with dimensions of 3 mm by 5 mm by 15 mm, a cylinder with a diameter of 2.6 m and height of 3.5 m, a rectangular prism with dimensions of 6 inches by 6 inches by 9 inches, and a cylinder with a radius of 3 yards and height of 10 yards. It also gives problems to find the height of a rectangular prism with given length, width, and volume, and to find the radius and height of cylinders with given dimensions and volumes.
04.08.10 applications of volume and surface area
The document contains 4 applications that use surface area and volume concepts:
1) Finding the height of an aquarium tank given its dimensions and volume.
2) Comparing the surface area and volume of a can and box to determine which holds more juice.
3) Calculating the surface area of a gift to see if a sheet of wrapping paper is large enough.
4) Determining which pizza sauce jar is a better buy based on price and volume.
04.07.10 more volume
The document provides instructions for calculating the volumes of different shapes, including a rectangular prism and cylinder. It asks the reader to first predict which solids have the greatest and least volumes without calculations. It then has the reader measure dimensions of different solids to calculate their actual volumes and rank them. Finally, it gives two word problems to find heights using given lengths, widths, radii and volumes.
Surface area of solids Review
The document discusses surface area of different solids including a square prism, rectangular prism, square pyramid, and cylinder. It provides information on calculating the surface area of different 3D shapes by identifying their nets and considering the areas of each face. The document serves as an introduction to finding the surface areas of common solids using their geometric properties.
Volume of spheres
The document describes an experiment to derive the formula for the volume of a sphere. Students fill hemispheres with rice and pour them into cylinders and cones to determine that the volume of a sphere is 4/3 πr^3, where r is the radius of the sphere. The experiment compares the volumes of hemispheres, cylinders, and cones to logically deduce the formula for a sphere's volume.
Surface area of circular solids
The document discusses calculating the surface areas of circular solids like cylinders and cones. It provides formulas for finding the surface area of a cylinder and a cone. For cylinders, the surface area is equal to the area of the two circular ends plus the area of the curved side surface. For cones, the surface area is the area of the circular base plus the area of the lateral curved surface, which can be calculated using the ratio of the sector area to the total circular area and the slant height.
m&m Statistics
The document discusses analyzing the number of M&Ms in different bags through collecting a sample from one bag, determining the population as the full bag, and calculating measures of center including the mean, mode, and median of the sample. It suggests creating a table and bar graph in Excel to display the data and compare results across bags. The document also notes that Mars determines official color distributions through consumer testing but individual bags may vary in color counts due to the filling process.
More from joannahstevens (20)
Solving equations with variables on both sides & translating words into symbols
Solving equations with variables on both sides & translating words into symbols
Simplifying expressions, solving, domain and range
Simplifying expressions, solving, domain and range
Inequalities
- 1. Inequalities August 19 and 23, 2010
- 2. Opener Give an example: {Negative numbers} is NOT closed under multiplication. Solve: x2= 324 State the domain and range of the function :
- 3. Section 1.6 - Inequalities < less than > greater than open dot ≤ less than or equal to ≥ greater than or equal to closed dot
- 4. Examples: Assume the domain is all real numbers 4x-7 > 1
- 5. Examples 3x-5 < 2
- 6. Examples x-4 > -1 or x-4 ≤ -6
- 7. Examples 4-3x > -2
- 8. Examples 3 ≤ x+4 < 5
- 9. Examples |x| > 3
- 10. Examples |x-1| < 4
- 12. Exit Slip 5 – 3x ≥ -4 4 – 3x < -2 |5x – 3| > 22 |x + 3| > -6