This document provides an overview of solving linear inequalities. It introduces inequality notation and properties, discusses multiplying and dividing by negative numbers, and provides examples of solving different types of linear inequalities. It also covers interval notation, graphing solutions to inequalities on number lines, and using interactive tools like Gizmos for additional practice with inequalities.
Power Point Presentation on a PAIR OF LINEAR EQUATION IN TWO VARIABLES, MATHS project...
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Pair Of Linear Equations In Two VariablesDeo Baran
PowerPoint Presentation of Learning Outcomes, Experiential content, Explanation Content, Hot Spot, Curiosity Questions, Mind Map, Question Bank of
Pair Of Linear Equations In Two Variables Class X
Power Point Presentation on a PAIR OF LINEAR EQUATION IN TWO VARIABLES, MATHS project...
Friends if you found this helpful please click the like button. and share it :) thanks for watching
Pair Of Linear Equations In Two VariablesDeo Baran
PowerPoint Presentation of Learning Outcomes, Experiential content, Explanation Content, Hot Spot, Curiosity Questions, Mind Map, Question Bank of
Pair Of Linear Equations In Two Variables Class X
This presentation explains Algebra in Mathematics. It includes: Introduction, Solution to Puzzle, Definition of terms, Rules in Algebra, Collecting Like Terms, Similar Terms, Expanding the Brackets, Nested Brackets, Multiplication of Algebraic Expressions of a Single Variable, Division of One Expression by another, Addition and Subtraction of Algebraic Fractions, Multiplication and Division of Algebraic Fractions, Factorisation of Algebraic Expression, Useful Products of Two Simple Factors, Examples, Trinomial Expression, Quadratic Expression as the Product of Two Simple Factors, Factorisation of Quadratic Expression ax2 + bx +c When a = 1, Factorisation of Quadratic Expression ax2 + bx +c When a ≠ 1 and Test for Simple Factors.
My talk about linear programming in NTU's APEX Club in NTU, Singapore in 2007. The club is for people who are keen on participating in ACM International Collegiate Programming Contests organized by IBM annually.
This presentation is intended to help the education students by giving an idea on how they will use technology in education specially in teaching mathematics.
This presentation explains Algebra in Mathematics. It includes: Introduction, Solution to Puzzle, Definition of terms, Rules in Algebra, Collecting Like Terms, Similar Terms, Expanding the Brackets, Nested Brackets, Multiplication of Algebraic Expressions of a Single Variable, Division of One Expression by another, Addition and Subtraction of Algebraic Fractions, Multiplication and Division of Algebraic Fractions, Factorisation of Algebraic Expression, Useful Products of Two Simple Factors, Examples, Trinomial Expression, Quadratic Expression as the Product of Two Simple Factors, Factorisation of Quadratic Expression ax2 + bx +c When a = 1, Factorisation of Quadratic Expression ax2 + bx +c When a ≠ 1 and Test for Simple Factors.
My talk about linear programming in NTU's APEX Club in NTU, Singapore in 2007. The club is for people who are keen on participating in ACM International Collegiate Programming Contests organized by IBM annually.
This presentation is intended to help the education students by giving an idea on how they will use technology in education specially in teaching mathematics.
You can use this presentation to introduce students in how to write linear equations given the slope and the y-intercept. This is the first case in writing linear equations.
This learner's module discusses and help the students about the topic Systems of Linear Inequalities. It includes definition, examples, applications of Systems of Linear Inequalities.
Std 10 computer chapter 9 Problems and Problem SolvingNuzhat Memon
Std 10 computer chapter 9 Problems and Problem Solving
Problem and Types of problem
Problem solving
Flowchart
Symbols of flowchart
Flowchart to calculate area of rectangle
Flowchart to calculate area and perimeter of circle
Flowchart to compute simple interest
Flowchart to find youngest student amongst two students
Flowchart to find youngest student amongst three students
Flowchart to find youngest student amongst any number of students
Flowchart to find sum of first 50 odd numbers
Flowchart to interchange or swap values of two variables with extra variable
Flowchart to interchange or swap values of two variables without extra variable
Advantage and disadvantage of flowchart
Algorithm
Advantage of flowchart
disadvantage of flowchart
Algorithm
Algorithm to find sum of numbers divisible by 11 in the range of 1 to 100
Algorithm to compute interest
Algorithm to find total weekly pay of employee
4. Solving an Inequality is just like solving an equation, almost…. Instead of an equal sign there is an inequality sign. If you multiply or divide by a negative number, you must flip the inequality sign. There is more than one number that is a part of the solution There can be 2 inequality signs in the problem.
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8. Definiton: A mathematical inequality is a statement that one expression is more than, more than or equal to, less than, or less than or equal to another. ***Not just simply equal to another Definiton: A linear inequality in one variable is an inequality in the form ax + b < c or ax + b < c, where a, b, and c are real numbers.
9. Example: Solve the inequality: 4x + 6 < 10 Solution: What does solving this inequality? It means that we are looking for the values of x that would make the value of 4x + 6 be less than 10 . Can you think of some? So, let’s find all of our solutions at once . We solve inequalities in much the same way as we do equations. What about 0, -18, 9, 3.6, -2.034, -1000, 2, … there’s a lot…too many to list Watch this video for a quick refresher!!!
10. This means that any value for x less than 1 would make the inequality 4x + 6 < 10 a true statement. For example, if x = -3 we have 4(-3)+6 = -6 < 10 However, if x = 1 we have 4(1) + 6 = 10 < 10. This is False ~ 10 is not less than 10! So we can only have answers that are less than 1 to make this a true statement. Think about what numbers are solutions to this problem. Solve the inequality: 4x + 6 < 10 (Any number less than 1.) How many solutions are there to this inequality? (An infinite amount)
11. Solve the inequality: -9 < x + 6 < 10 This inequality can be solved 2 different ways, but the answer will be the same wither way.
12. More Examples: a) Solve: 3x – 10 > 9x + 5 Solution: b) Solve: -7 < 5x – 3 < 2 This is a compound inequality *****Answer will also be a compound inequality Solution:
13. c) Suppose you were working in the finance department of a factory that produces widgets. The total cost of producing widgets depends on the number of them produced. In order to make one batch, your factory has a fixed (one-time) cost of $1500, and it costs $35 to produce each widget. 1. Write an equation that represents the total cost, C , of producing w widgets. 2. Suppose the factory has a budget of at most $32,000 to spend on making a batch of widgets. What is the range of the number of widgets your factory can produce?
14. Solutions: 1. If C is the total cost of producing a batch, and w is the number of widgets, then the equation that represents their relationship is C = 35w + 1500 2. Since the budget is at most $32,000, we want to find the range of values for w such that 35w + 1500 < 32000. We include 32,000 in our inequality, since the budget includes 32,000 as a possibility. We then proceed to solve the inequality. So, 6100/7 is approximately 871.43. Since w needs to be less than or equal to 6100/7, and it MUST be a whole number (you wouldn’t want a fraction of a widget) we will have to round to 871. So, the answer to our finance question is that our factory can produce at most 871 widgets to meet the budget. (We could have also said this by saying that our factory can produce no more than 871 widgets to meet the budget.)
15. d) Suppose you’re making a slow-cooking stew, and the recipe states that the temperature of your stew must remain strictly between 45ºC and 55ºC for 3 hours in order to cook properly. What would this range be in degrees Fahrenheit? Solution : The relationship between Celsius and Fahrenheit is Since we want to find F such that 45 < C < 55, the inequality we want to solve is So, we find that the stew must remain strictly between 113ºF and 131ºF for 3 hours to cook properly.
16. You are in college. Your final math class grade consists of 4 test grades. You’ve earned a 78%, 88%, and 92%, respectively, on your first three tests this semester. Your parents tell you must get a final grade between 82% and 89%. What range of scores could you earn on your fourth test in order to have a class average between 82% and 89%? You know that to find the average of a set of numbers, you add them and divided by the number of items you added. You know 3 of the 4 test scores here. For any unknown, we can call it x. We want our answer to be a range of scores, all the scores, that give us the average we are looking for, The lowest average we want is 82% and the highest average is 89% Go to next slide to finish problem
17. -258 -258 -258 This means that the lowest score you can get on your 4 th test is a 70% and the high score of 98% on the 4 th test will give you an average of an 89%. Any score between the 70% and 98% will give you an average between 82% and 89%
18. Interval Notation There is another way to write the solutions to linear inequalities. It is called interval notation. Here is a summary: < or > (without the equal sign) is shown by using ( < or > is shown by using [ (infinity) is used when there is a single sided inequality. So, in the example above 113 < F < 131, we would write in interval notation as [113,131). More examples: 5< x < 9 is written as : (5, 9] 6 < x < 25 is written as: [6, 25) Now, a single sided inequality uses the infinity symbol: x > 5 is written as : (5, ). We use the parenthesis at 5 because there is no equals sign under the greater than sign. The infinity symbol always uses the parenthesis. x < 15 is written as : ( , 15].
19. Interval Notation Write the following solution in interval notation: Notice the solid circle gets the [ because - 1 is included, and the open circle gets the ) because 2 is not included. Interactive Example of Interval Notation and Set Notation (Double-click on endpoints to change between included and excluded endpoints)
21. More on Interval Notation Interval Notation Examples, Interval Notation, Graphing on Number Line Practice Problems and Answers How to Solve Multistep Inequalities How to Solve Inequalities and Graph on a Number Line Compound Inequalities Solving and Graphing Compound Inequalities
22. Gizmos Gizmo: Solving Inequalities using Multiplication and Division Gizmo: Compound Inequalities Gizmo: Solving Inequalities using Addition and Subtraction Gizmo: Solving Inequalities using Multiplication and Division
23. Example 1: -45 -45 -10m -10m 140 140 Remember this problem? We did this as an equation in Topic 1 Notes. Nothing changes, except the sign.
24. Practice Examples Example 2: Example 3: Solve the equation 3p + 2 < 0 . Solve the equation - 7m – 1 > 0. Solutions on next slide. Solve these on your own first. Example 4: Solve the equation 14z – 28 > 0.
25. Practice Examples Answers Example 2: Example 3: Solve the equation 3p + 2 < 0 . Solve the equation -7m – 1 > 0. Example 4: Solve the equation 14z – 28 > 0. Notice the sign flips because you divide by a – 7.
26. More Practice Examples Example 5: Example 6: Solve the equation . Solve the equation . Solve these on your own first. Solutions on next slide.
27. More Practice Examples - Answers Example 5: Example 6: Solve the equation . Solve the equation .
28. Graphing Answers to Inequalities When graphing and inequality: < shade left (toward negatives) < shade left (toward negatives) > shade right (toward positives) > shade right (toward positives) When graphing and inequality: < open circle > open circle < closed circle > closed circle When the variable is to the left of the inequality sign: