Lesson 3: Solving Linear Inequalities Remediation Notes

Solving Linear Inequalities Some real-life problems that involve mathematics are based in determining a range of values that satisfy the expressions involved. For these types of problems, we need to set up inequalities, as opposed to equations, to properly solve them.

almost…. Solving an Inequality is just like solving an equation, 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 We will begin by reminding ourselves of formal mathematical notation and the properties of inequalities. Greater than > Ex: 12 > 6 Greater than or equal to > Ex: 17 > 16 Less than < Ex: 9 < 13 Less than or equal to < Ex: -15 < 15

Multiplying by Negatives  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. Well, if we consider the inequality 3 > 1 , how do the two numbers’ opposites relate? In other words, how do -1 and -3 relate? Which one is larger? We know that -1 is larger than -3; i.e. -3 < -1. Notice that the inequality changes direction, because we are now looking at the left side of zero on the number line.

***Not just simply equal to another 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 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. Definiton: Image from: http://commons.wikimedia.org/wiki/File:Chalkandboard.jpg

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? What about 0, -18, 9, 3.6, -2.034, -1000, 2, …there’s a lot…too many to list So, let’s find all of our solutions at once. We solve inequalities in much the same way as we do equations.

Solve the inequality: 4x + 6 < 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. (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.

Go to next slide to finish problem 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 4th test is a 70% and the high score of 98% on the 4th 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].

Interactive Example of Interval Notation and Set 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:

How to Solve Inequalities and Graph on a Number Line How to Solve Multistep Inequalities Interval Notation Solving and Graphing Compound Inequalities Examples, Interval Notation, Graphing on Number Line More on Interval Notation Practice Problems and Answers

Gizmos Gizmo: Solving Inequalities using Addition and Subtraction Gizmo: Solving Inequalities using Multiplication and Division Gizmo: Solving Inequalities using Multiplication and Division Gizmo: Compound Inequalities

Graphing Answers to Inequalities When the variable is to the left of the inequality sign: When graphing and inequality: < open circle > open circle < closed circle > closed circle When graphing and inequality: < shade left (toward negatives) > shade right (toward positives)