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Linear Inequalities Solving Linear Inequalities

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7/2/2013 Linear Inequalities 2 2 Meanings and Solutions What does f(x) < g(x) mean? f(x) < g(x) is an inequality It says f(x) is less than g(x) (algebraically smaller than g(x)) … for certain values of x This may be TRUE for some values of x and FALSE for others Inequalities

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7/2/2013 Linear Inequalities 3 3 Meanings and Solutions Other inequality forms include ≤, >, and ≥ Now what do we do with inequalities ? We solve them, i.e. find their solutions Convert to equivalent inequality, i.e. one with same solutions Inequalities

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7/2/2013 Linear Inequalities 4 4 Finding Solutions What is a solution for an inequality? A solution for f(x) < g(x) is a value of x that makes the inequality TRUE Any x that makes the inequality false is not a solution Inequalities

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7/2/2013 Linear Inequalities 5 5 Finding Solutions What is the solution set for an inequality? The solution set for the inequality is the set of all solutions The solution set might be in several discrete pieces The solution set might be empty Inequalities

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7/2/2013 Linear Inequalities 6 6 Examples 1. 3(2x + 3) < 4x + 1 2. – x(x – 4) ≥ x – 4 Inequalities TRUE for all x < –4, FALSE otherwise TRUE for –1 ≤ x ≤ 4, FALSE otherwise Question: What are the solution sets for the above ?

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7/2/2013 Linear Inequalities 7 7 Examples 3. 2y – 4 ≤ x Inequalities TRUE for all (x,y) where y ≤ x + 2, 1 2 FALSE otherwise Question: What are the solution sets for the above ? 4. 3 > 4 + 2x, for x > 0 Logically FALSE WHY ?

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7/2/2013 Linear Inequalities 8 8 Examples 5. 3 + 4 ≤ 7 Inequalities Question: What are the solution sets for the above ? 6. (x – 2)2 + 3 ≥ 1, for x ≥ 1 Logically TRUE WHY ? Logically TRUE WHY ?

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7/2/2013 Linear Inequalities 9 9 Addition Rule: If a > b then a + c > b + c for any real c Examples If 7 > 3 then 7 + 4 > 3 + 4 If 7 > 3 then 7 – 9 > 3 – 9 If 2x + 5 > 3 then (2x + 5) – 5 > 3 – 5 Inequalities: Rules of the Road OR 2x > -2 … an equivalent inequality

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7/2/2013 Linear Inequalities 10 Multiplication Rule 1: If a > b and c > 0 then a c > bc Examples If 7 > 3 and 5 > 0 then 7(5) > 3(5) If 2x + 6 > 8 then ½(2x + 6) > ½(8) Inequalities: Rules of the Road … an equivalent inequality OR x + 3 > 4

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7/2/2013 Linear Inequalities 11 Multiplication Rule 2: If a > b and c < 0 then a c < bc Examples If 7 > 3 and -5 < 0 then 7(-5) < 3(-5) If 2x + 6 > 8 then -½(2x + 6) < -½(8) Inequalities: Rules of the Road … an equivalent inequality OR -x – 3 < -4 Question:What if c = 0 ?

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7/2/2013 Linear Inequalities 12 Definition A linear inequality in one variable is one that can be written a x + b > 0 in standard form, with a ≠ 0 Linear Inequalities … also includes forms with ≥, <, ≤

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7/2/2013 Linear Inequalities 13 Examples 1. 2x – 5 < 1 2. 3 – 4x ≥ 5 3. 2x + 1 ≤ 3x + 7 Linear Inequalities Question: Can each of these be written in standard form ? 2x – 6 < 0 -4x – 2 ≥ 0 -x – 6 ≤ 0 4x + 2 ≤ 0 x + 6 ≥ 0

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7/2/2013 Linear Inequalities 14 Definition A solution for a linear inequality in one variable is a value of the variable that makes the inequality TRUE The set of all solutions for an inequality is the solution set for the inequality Solutions of Linear Inequalities

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7/2/2013 Linear Inequalities 15 Example 2x – 5 < 1 Some solutions are: 2, 2.5, 1, 0, -5.4, … Solution Set Notation Set notation: { x x < 3 } Interval notation: ( – , 3) Solutions of Linear Inequalities Question: Does this interval include 3 ? How many solutions?

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7/2/2013 Linear Inequalities 16 Example 1: Analytical Method Solve: Solving: Solution Set is: Inequalities 3 – 2x ≤ 5 – 2x ≤ 5 – 3 = 2 x ≥ – 1 or [ – 1, ) { x | x ≥ – 1 }

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7/2/2013 Linear Inequalities 17 Example 1: Graphical Method Solve: Solutions: Solution Set is: Inequalities 3 – 2x ≤ 5 or [ – 1, ) y x y 2 = 5 y 1 = 3 – 2x (-1, 5) y 1 = y 2 0135-3-5 –– [ { x | x ≥ – 1 } y 2 = 5 y 1 = 3 – 2x

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7/2/2013 Linear Inequalities 18 Example 2: Analytical Method Solve: Solving: Solution Set is: Inequalities 5x – 1 < 2x + 11 5x – 2x < 11 + 1 3x < 12 x < 4 or ( – , 4 ) { x | x < 4 }

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7/2/2013 Linear Inequalities 19 Example 2: Graphical Method Solve: Graphically: Solution Set is: y x Inequalities 5x – 1 < 2x + 11 y 2 = 2x + 11 (4, 19) or ( – , 4 ) y 1 = 5x – 1 y 1 < y 2 4 { x | x < 4 } 024-2-4 –– ) y 1 = 5x – 1 y 2 = 2x + 11

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7/2/2013 Linear Inequalities 20 Notes on Notation Symbols Sometimes the same symbols are used to mean different things – just as words in English can have different meanings Points and Intervals The point (2, 3) and the open interval (2, 3) use the same notation The difference is determined by context

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7/2/2013 Linear Inequalities 21 Notes on Notation Points of Confusion Do not write (3, 7) or [3, 7] when you mean { 3, 7 } Never write [– , 7 ], [ , 7 ), (3, ], or [3, ] WHY ?

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7/2/2013 Linear Inequalities 22 Rewrite inequality in form y ) Example 5x – 2x – 1 – 11 < 0 3x – 12 < 0 Let y = 3x – 12 y = 3x – 12 = 0 for x = 4 (4, 0) y x Horizontal Intercept Method y = 3x – 12 y = 3x – 12 < 0 5x – 1 < 2x + 11 < 0

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7/2/2013 Linear Inequalities 23 Rewrite inequality in form y ) Example y = 3x – 12 = 0 for x = 4 For y < 0, we have x < 4 Solution set: { x x < 4 } or (– , 4) (4, 0) y x y = 3x – 12 y = 3x – 12 < 0 Horizontal Intercept Method – 024-2-4 ) 5x – 1 < 2x + 11

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7/2/2013 Linear Inequalities 24 Recall: Basic Absolute Value Facts 1. x ≥ 0 for all real x 2. x = –x for all real x Absolute Values in Inequalities a a = –a, for a < 0 a, for a ≥ 0

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7/2/2013 Linear Inequalities 25 Basic Absolute Value Facts 3. If x < b then –b < x < b 4. If x > b > 0 x Absolute Values in Inequalities WHY ? 0 x b –b –xx –x x WHY ? 0 b –b x x –x –x then either x > b or x < -b

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7/2/2013 Linear Inequalities 26 Examples 1. │ 3 │ < 7 then –7 < 3 < 7 2. │–3│ < 7 then –7 < –3 < 7 Absolute Values in Inequalities │3││3│0 3 7 –7 7 –3 │–3│ 0

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7/2/2013 Linear Inequalities 27 Examples 3. │ –7 │ > 3 Rule: Replace absolute value with each of the two forms indicated in the definition │–7│ Absolute Values in Inequalities 3 –3 –7 7 0 then either –7 > 3 or –7 < –3

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7/2/2013 Linear Inequalities 28 Examples 4. If │x + 2 │< 9 Absolute Values in Inequalities │x + 2 │ xx –9 x + 2 9 –1170x + 2 then –9 < x + 2 < 9 and –11 < x < 7

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7/2/2013 Linear Inequalities 29 Examples 5. If x + 2 > 7 > 0 Absolute Values in Inequalities then either x > 5 or x < –9 │x + 2 │ x + 2 xx –9 5–77x + 2 0 –│x + 2│

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7/2/2013 Linear Inequalities 30 Compound Inequalities More Than One Inequality 1. Find the solution set for x ≥ –3 and x < 7 Rewriting: –3 ≤ x and x < 7 Solution set is { x –3 ≤ x < 7 } OR [ –3, 7 ) OR compounding: –3 ≤ x < 7 – 0135-3-57911-7-9 [ )

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7/2/2013 Linear Inequalities 31 Compound Inequalities More Than One Inequality 2. Find the solution set for x < –3 OR x ≥ 7 Can’t rewrite as 7 ≤ x < –3 Solution set is { x x < –3 } { x x ≥ 7 } WHY ? OR ( – , –3) [ 7, ) – 0135-3-57911-7-9 ) [

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7/2/2013 Linear Inequalities 32 Compound Inequalities Example Find the solution set for 8 – 4t < – 15 < – 7 4 23 4 >> t Note: Clear the fractions: Simplify: –23 < –4t < 7 – 2 – t 5 3 4 < < 3 4 – 3 4 20 ( ) < 2 – t 5 20 ( ) < 3 4 20 ( ) Inequalities reversed

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7/2/2013 Linear Inequalities 33 Compound Inequalities Example Find the solution set for – 7 4 23 4 >> t – 2 – t 5 3 4 < < 3 4 Solution set OR Rewriting – 7 4 23 4 << t – 7 4 4 << t { t | } 4, – 7 4 () – 024-2-46810-6 ) (

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7/2/2013 Linear Inequalities 34 Compound Inequalities Example Find the solution set for 10x + 2(x – 4) < 12x – 10 –8 < –10 Simplifying: 10x + 2x – 8 < 12x – 10 A logically FALSE statement ! There is no value of x … that will make this true !

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7/2/2013 Linear Inequalities 35 Compound Inequalities Example Find the solution set for 10x + 2(x – 4) < 12x – 10 There is no value of x … that will make this true ! … OR just { } Note: WHY ? We say the solution set is EMPTY ! We write: Solution set is O O ≠ O { }

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7/2/2013 Linear Inequalities 36 Think about it !

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