Module 1 ~ Topic 1 Solving Inequalities <ul><li>Table of Contents </li></ul><ul><li>Slides 3-4: Introduction to Inequalities </li></ul><ul><li>Slides 5-6: Notation and Properties </li></ul><ul><li>Slide 7: Multiplying/Dividing by a Negative Number </li></ul><ul><li>Slides 9-15: Examples </li></ul><ul><li>Slides 16 – 17: Compound Inequalities </li></ul><ul><li>Slides 18-20: Interval Notation </li></ul><ul><li>Slides 23-27: Examples </li></ul><ul><li>Slides 28-30: Graphing Solutions </li></ul><ul><li>Audio/Video and Interactive Sites </li></ul><ul><li>Slide 2: Graphing Calculator Use Guide </li></ul><ul><li>Slide 9: Video </li></ul><ul><li>Slide 19: Interactive Site </li></ul><ul><li>Slide 21: Videos </li></ul><ul><li>Slide 22: Gizmos </li></ul>
Special Instructions <ul><li>This module includes graphing calculator work. </li></ul><ul><li>Refer to the website, TI-83/84 calculator instructions , for resources and instructions on how to use your calculator to obtain the results we do throughout the lessons. I suggest that you bookmark this site if you haven’t done so already. </li></ul><ul><li>Take careful notes and following along with every example in each lesson. I encourage you to ask me questions and think deeply as you’re studying these concepts! </li></ul>
Topic #2: Solving Linear Inequalities <ul><li>Some real-life problems that involve mathematics are based in determining a range of values that satisfy the expressions involved. </li></ul><ul><li>For these types of problems, we need to set up inequalities , as opposed to equations, to properly solve them. </li></ul>
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.
Notation <ul><li>We will begin by reminding ourselves of formal mathematical notation and the properties of inequalities. </li></ul><ul><li>Greater than > Ex: 12 > 6 </li></ul><ul><li>Greater than or equal to > Ex: 17 > 16 </li></ul><ul><li>Less than < Ex: 9 < 13 </li></ul><ul><li>Less than or equal to < Ex: -15 < 15 </li></ul>
Properties <ul><li>Let a , b , c , and d be real numbers. (The following apply to all inequality types (> ,< , > , and < ) analogously.) </li></ul><ul><li>1. If a < b and b < c then a < c. ( Transitive Property ) </li></ul><ul><li>2. If a < b and c < d then a + c < b + d. ( Addition Property ) </li></ul><ul><li>3. If a < b, then a + c < b + c. ( Addition of a Constant ) </li></ul><ul><li>4. a. If a < b and c is positive, then ac < ab. In other words, the inequality is preserved . </li></ul><ul><li>b. If a < b and c is negative, then ac > bc. In other words, the inequality reverses direction. ( Multiplication by a Nonzero Constant ) </li></ul>
Multiplying by Negatives <ul><li>Note: Let’s take a moment to understand why, when we multiply both sides of an inequality by a negative number, the inequality reverses direction . </li></ul><ul><li>Well, if we consider the inequality 3 > 1 , how do the two </li></ul><ul><li>numbers’ opposites relate? </li></ul><ul><li>In other words, how do - 1 and - 3 relate? </li></ul><ul><li>Which one is larger? </li></ul><ul><li>We know that - 1 is larger than - 3; </li></ul><ul><li>i.e. - 3 < - 1. </li></ul><ul><li>Notice that the inequality changes direction, because we are now looking at the left side of zero on the number line . </li></ul>
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.
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!!!
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)
Solve the inequality: -9 < x + 6 < 10 This inequality can be solved 2 different ways, but the answer will be the same wither way.
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:
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?
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.)
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.
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
-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%
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].
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)
Interval Notation Write the following solutions in interval notation:
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
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
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.
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.
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.
More Practice Examples Example 5: Example 6: Solve the equation . Solve the equation . Solve these on your own first. Solutions on next slide.
More Practice Examples - Answers Example 5: Example 6: Solve the equation . Solve the equation .
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:
Lets graph some of the answers we have already found: