Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
SlideShare a Scribd company logo

Em06 iav

•
•
N
nahomyitbarek

1) The document discusses inequalities involving real numbers. It introduces basic rules for how inequalities are affected by addition, multiplication, taking squares, and taking reciprocals. 2) Several examples are provided to demonstrate how to solve various inequalities by applying these rules. For instance, it is shown that the inequality 4x + 7 < 3 is equivalent to x < -1. 3) Graphs are used to confirm the solutions obtained from manipulating the inequalities. The key results are that addition preserves inequalities while multiplication preserves or reverses them depending on whether the number being multiplied is positive or negative.

1 of 9
Download to read offline
ELEMENTARY MATHEMATICS W W L CHEN and X T DUONG c W W L Chen, X T Duong and Macquarie University, 1999. This work is available free, in the hope that it will be useful. Any part of this work may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, with or without permission from the authors. Chapter 6 INEQUALITIES AND ABSOLUTE VALUES 6.1. Some Simple Inequalities Basic inequalities concerning the real numbers are simple, provided that we exercise due care. We begin by studying the effect of addition and multiplication on inequalities. ADDITION AND MULTIPLICATION RULES. Suppose that a, b ∈ R and a < b. Then (a) for every c ∈ R, we have a + c < b + c; (b) for every c ∈ R satisfying c > 0, we have ac < bc; and (c) for every c ∈ R satisfying c < 0, we have ac > bc. In other words, addition by a real number c preserves the inequality. On the other hand, multipli- cation by a real number c preserves the inequality if c > 0 and reverses the inequality if c < 0. Remark. We can deduce some special rules for positive real numbers. Suppose that a, b, c, d ∈ R are all positive. If a < b and c < d, then ac < bd. To see this, note simply that by part (b) above, we have ac < bc and bc < bd. SQUARE AND RECIPROCAL RULES. Suppose that a, b ∈ R and 0 < a < b. Then (a) a2 < b2 ; and (b) a−1 > b−1 . Proof. Part (a) is a special case of our Remark if we take c = a and d = b. To show part (b), note that 1 1 b−a a−1 − b−1 = − = > 0. ♣ a b ab † This chapter was written at Macquarie University in 1999.
6–2 W W L Chen and X T Duong : Elementary Mathematics CAUCHY’S INEQUALITY. For every a, b ∈ R, we have a2 + b2 ≥ 2ab. Furthermore, equality holds precisely when a = b. Proof. Simply note that a2 + b2 − 2ab = a2 − 2ab + b2 = (a − b)2 ≥ 0, and that equality holds precisely when a − b = 0. ♣ We now use some of the above rules to solve inequalities. We shall illustrate the ideas by considering a few examples in some detail. Example 6.1.1. Consider the inequality 4x + 7 < 3. Using the Addition rule and adding −7 to both sides, we obtain 4x < −4. Using one of the Multiplication rules and multiplying both sides by the positive real number 1/4, we obtain x < −1. We have shown that 4x + 7 < 3 =⇒ x < −1. Suppose now that x < −1. Using one of the Multiplication rules and multiplying both sides by the positive real number 4, we obtain 4x < −4. Using the Addition rule and adding 7 to both sides, we obtain 4x + 7 < 3. Combining this with our earlier observation, we have now shown that 4x + 7 < 3 ⇐⇒ x < −1. We can confirm our conclusion by drawing a graph of the line y = 4x + 7 and observing that the part of the line below the horizontal line y = 3 corresponds to x < −1 on the x-axis. y 7 6 y = 4x + 7 5 4 y =3 3 2 1 x -2 -1 1 -1
Chapter 6 : Inequalities and Absolute Values 6–3 Example 6.1.2. Consider the inequality −5x + 4 > −1. Using one of the Multiplication rules and multiplying both sides by the negative real number −1, we obtain 5x − 4 < 1. Using the Addition rule and adding 4 to both sides, we obtain 5x < 5. Using one of the Multiplication rules and multiplying both sides by the positive real number 1/5, we obtain x < 1. We have shown that −5x + 4 > −1 =⇒ x < 1. Suppose now that x < 1. Using one of the Multiplication rules and multiplying both sides by the positive real number 5, we obtain 5x < 5. Using the Addition rule and adding −4 to both sides, we obtain 5x − 4 < 1. Using one of the Multiplication rules and multiplying both sides by the negative real number −1, we obtain −5x + 4 > −1. Combining this with our earlier observation, we have now shown that −5x + 4 > −1 ⇐⇒ x < 1. We can confirm our conclusion by drawing a graph of the line y = −5x + 4 and observing that the part of the line above the horizontal line y = −1 corresponds to x < 1 on the x-axis. y 6 5 4 3 y = -5x + 4 2 1 x -1 1 2 -1 y = -1 -2 -3 Example 6.1.3. Consider the inequality x2 ≤ a2 , where a > 0 is fixed. Clearly x = ±a are the only solutions of the equation x2 = a2 . So let us consider the inequality x2 < a2 . Observe first of all that the inequality is satisfied when x = 0. On the other hand, if 0 < x < a, then the Square rule gives x2 < a2 . However, if −a < x < 0, then using one of the Multiplication rules and multiplying all sides by the negative real number −1, we obtain a > −x > 0. It follows from the Square rule that (−x)2 < a2 , so that x2 < a2 . We have now shown that −a ≤ x ≤ a =⇒ x2 ≤ a2 . Suppose now that x > a. Then it follows from the Square rule that x2 > a2 . On the other hand, suppose that x < −a. Using one of the Multiplication rules and multiplying both sides by the negative real number −1, we obtain −x > a. It follows from the Square rule that (−x)2 > a2 , so that x2 > a2 . We have now shown that x < −a or x > a =⇒ x2 > a2 . It now follows that −a ≤ x ≤ a ⇐⇒ x2 ≤ a2 .
6–4 W W L Chen and X T Duong : Elementary Mathematics We can confirm our conclusion by drawing a graph of the parabola y = x2 and observing that the part of the parabola on or below the horizontal line y = a2 corresponds to −a ≤ x ≤ a on the x-axis. y a2 y = x2 x -a a Example 6.1.4. Consider the inequality x2 − 4x + 3 ≤ 0. We can write x2 − 4x + 3 = x2 − 4x + 4 − 1 = (x − 2)2 − 1, so that the inequality is equivalent to (x − 2)2 − 1 ≤ 0, which in turn is equivalent to the inequality (x − 2)2 ≤ 1, in view of the Addition rule. Now write u = x − 2. Then it follows from Example 6.1.3 that −1 ≤ u ≤ 1 ⇐⇒ u2 ≤ 1. Hence −1 ≤ x − 2 ≤ 1 ⇐⇒ (x − 2)2 ≤ 1. Using the addition rule on the inequalities on the left hand side, and using our earlier observation, we conclude that 1≤x≤3 ⇐⇒ x2 − 4x + 3 ≤ 0. We can confirm our conclusion by drawing a graph of the parabola y = x2 − 4x + 3 and observing that the part of the parabola on or below the horizontal line y = 0 corresponds to 1 ≤ x ≤ 3 on the x-axis. y 3 2 y = x2 − 4x + 3 1 x 1 2 3 4 -1 Example 6.1.5. Consider the inequality 1 < 2. x Clearly we cannot have x = 0, as 1/0 is meaningless. We have two cases: (1) Suppose that x > 0. Using one of the Multiplication rules and multiplying both sides by the positive real number x, we obtain the inequality 1 < 2x. Multiplying both sides by the positive real number 1/2, we obtain 1/2 < x. Suppose now that 1/2 < x. Using one of the Multiplication rules
Chapter 6 : Inequalities and Absolute Values 6–5 and multiplying both sides by the positive real number 2/x, we obtain the original inequality. We have therefore shown that for x > 0, we have 1 1 x> ⇐⇒ < 2. 2 x (2) Try to use one of the Multiplication rules to show that 1 x<0 ⇐⇒ < 0. x The result is obvious, but the proof is slightly tricky. Combining the two parts, we conclude that 1 1 x<0 or x > ⇐⇒ < 2. 2 x We can confirm our conclusion by drawing a graph of the hyperbola y = 1/x and observing that the part of the hyperbola below the horizontal line y = 2 corresponds to x < 0 together with x > 1/2 on the x-axis. y y = 1/x y=2 2 1 x 1 For the remaining examples in this section, we shall use M to denote an application of one of the Multiplication rules, A to denote an application of the Addition rule and S to denote an application of the Square rule. Example 6.1.6. Consider the inequality x+4 < 3. 2x Clearly we cannot have x = 0. We therefore have two cases: (1) Suppose that x > 0. We have x+4 M A M 4 <3 ⇐⇒ x + 4 < 6x ⇐⇒ 4 < 5x ⇐⇒ < x. 2x 5
6–6 W W L Chen and X T Duong : Elementary Mathematics (2) Suppose that x < 0. We have x+4 M A M 4 <3 ⇐⇒ x + 4 > 6x ⇐⇒ 4 > 5x ⇐⇒ > x. 2x 5 Since the rightmost inequality is always satisfied when x < 0, it follows that the leftmost inequality is always satisfied when x < 0. Combining the two parts, we conclude that 4 x+4 x<0 or x > ⇐⇒ < 3. 5 2x Example 6.1.7. Consider the inequality x2 − 3x + 4 ≤ 1. x+1 Clearly we cannot have x = −1. We therefore have two cases: (1) Suppose that x > −1, so that x + 1 > 0. Then x2 − 3x + 4 M A ≤1 ⇐⇒ x2 − 3x + 4 ≤ x + 1 ⇐⇒ x2 − 4x + 3 ≤ 0. x+1 Recall from Example 6.1.4 that 1≤x≤3 ⇐⇒ x2 − 4x + 3 ≤ 0. (2) Suppose that x < −1, so that x + 1 < 0. Then x2 − 3x + 4 M A ≤1 ⇐⇒ x2 − 3x + 4 ≥ x + 1 ⇐⇒ x2 − 4x + 3 ≥ 0. x+1 It can be deduced from Example 6.1.4 that x≤1 or x ≥ 3 ⇐⇒ x2 − 4x + 3 ≥ 0. Combining the two parts, we conclude that x2 − 3x + 4 x < −1 or 1 ≤ x ≤ 3 ⇐⇒ ≤ 1. x+1 Example 6.1.8. Suppose that a, b ∈ R are non-negative. We shall prove that a+b √ ≥ ab. 2 Suppose on the contrary that this is not true. Then a+b √ S (a + b)2 M < ab =⇒ < ab =⇒ (a + b)2 < 4ab 2 4 A =⇒ (a − b)2 = (a + b)2 − 4ab < 0. But the last inequality is absurd. Example 6.1.9. Suppose that x ∈ R and x > 0. We shall prove that x + x−1 ≥ 2. Suppose on the contrary that this is not true. Then x + x−1 < 2 M A =⇒ x2 + 1 < 2x =⇒ (x − 1)2 = x2 − 2x + 1 < 0. But the last inequality is absurd.
Chapter 6 : Inequalities and Absolute Values 6–7 Example 6.1.10. Suppose that x ∈ R. We shall prove that x2 − 4x + 3 ≥ −1. Suppose on the contrary that this is not true. Then A x2 − 4x + 3 < −1 =⇒ (x − 2)2 = x2 − 4x + 4 < 0. But the last inequality is absurd. 6.2. Absolute Values Definition. For every a ∈ R, the absolute value |a| of a is a non-negative real number satisfying a if a ≥ 0; |a| = −a if a < 0. Remark. If we place the number a on the real number line, then the absolute value |a| represents the distance of a from the origin 0. PROPERTIES OF ABSOLUTE VALUES. For every a, b ∈ R, we have (a) |a| ≥ 0; (b) |a| ≥ a; (c) |a|2 = a2 ; (d) |ab| = |a||b|; and (e) |a + b| ≤ |a| + |b|. The graph of the function y = |x| is given below, where a ∈ R is a non-negative real number. y y = |x| a x -a +a The following is easily seen from the graph. RULE CONCERNING REMOVAL OF ABSOLUTE VALUES. Suppose that a ∈ R is non- negative. Then |x| ≤ a if and only if − a ≤ x ≤ +a, and |x| < a if and only if − a < x < +a. Example 6.2.1. The equation |x| = 4 has two solutions x = ±4.
6–8 W W L Chen and X T Duong : Elementary Mathematics Example 6.2.2. The equation |2x + 1| = 5 has two solutions, one satisfying 2x + 1 = 5 and the other satisfying 2x + 1 = −5. Hence x = 2 or x = −3. Example 6.2.3. The inequality |x| < 5 is satisfied precisely when −5 < x < 5. Example 6.2.4. The inequality |2x + 1| ≤ 9 is satisfied precisely when −9 ≤ 2x + 1 ≤ 9; in other words, when −5 ≤ x ≤ 4. √ Example 6.2.5. The equation x2 + 4x + 13 = x − 1 is satisfied only if the right hand side is non- negative, so that we must have x ≥ 1. Squaring both sides, we have x2 + 4x + 13 = (x − 1)2 = x2 − 2x + 1, so that 6x + 12 = 0, giving x = −2. Hence the equation has no real solution x. Example 6.2.6. Consider the inequalities 3 < |x+4| < 7. Note first of all that the inequality |x+4| < 7 holds precisely when −7 < x + 4 < 7; in other words, when −11 < x < 3. On the other hand, the inequality |x + 4| ≤ 3 holds precisely when −3 ≤ x + 4 ≤ 3; in other words, precisely when −7 ≤ x ≤ −1. Hence the inequality |x + 4| > 3 holds precisely when x < −7 or x > −1. It follows that the original inequalities hold precisely when −11 < x < −7 or −1 < x < 3. We can confirm our conclusion by drawing a graph of the function y = |x + 4| and observing that the part of the graph between the horizontal lines y = 3 and y = 7 corresponds to −11 < x < −7 together with −1 < x < 3 on the x-axis. y y = |x + 4| y=7 y=3 x -11 -7 -4 -1 3 Problems for Chapter 6 1. Suppose that α and β are two positive real numbers. The number 1 (α + β) is called the arithmetic √ 2 mean of α and β, while the number αβ is called the geometric mean of α and β. √ a) Prove that αβ ≤ 1 (α + β); in other words, the geometric mean never exceeds the arithmetic 2 mean. b) Show that equality holds in part (a) precisely when α = β. [Remark: The famous arithmetic-mean-geometric-mean theorem states that for any k positive √ integers α1 , . . . , αk , we have k α1 . . . αk ≤ k (α1 + . . . + αk ), and that equality holds precisely when 1 α1 = . . . = αk . The proof is rather complicated when k > 2, and is beyond the scope of our present discussion.] 2. For each of the following inequalities, find all real values of x satisfying the inequality: a) 2x + 4 < 6 b) 5 − 3x > 11 c) 7x + 9 > −5 d) 4x + 4 < 28 e) 2x + 5 < 3 f) 4 − 6x ≥ 10 3. Determine all real values of x for which the inequalities 5 < 2x + 7 ≤ 13 hold.
Chapter 6 : Inequalities and Absolute Values 6–9 4. For each of the following inequalities, determine all real values of x for which the inequality holds: a) 6 + x − x2 ≥ 0 b) x2 − 1 > 0 c) x2 − 4 ≤ 0 d) 2 − x − x2 ≥ 0 e) x2 + 2x + 1 ≤ 0 f) (2x + 3)2 ≤ 4 g) (3x − 1)2 > 9 5. For each of the following inequalities, determine all real values of x for which the inequality holds, taking care to distinguish the two cases x > 0 and x < 0, and explain each step of your argument by quoting the relevant rules concerning inequalities: x+4 1 1 a) <3 b) <3 c) −2 < < 3 2x x x 6. For each of the following inequalities, determine all real values of x for which the inequality holds, taking care to distinguish two cases, and explain each step of your argument by quoting the relevant rules concerning inequalities: 2x + 3 4x − 2 4x − 2 a) <1 b) ≥2 c) 2 ≤ <3 3x + 1 x+4 x+4 x2 + 4x − 7 7. Show that ≥ 2 precisely when −5 ≤ x < −4 or x ≥ 3. x+4 8. For each of the following inequalities, determine all real values of x for which the inequality holds: x2 − 5x + 3 x2 − 4x + 1 a) ≤1 b) ≤1 x−2 x−3 9. Find all solutions of the inequality |x + 2| < 6, and confirm your answer by drawing a suitable picture. 10. For each of the following inequalities, determine all real values of x for which the inequality holds: a) 1 < |3x − 5| ≤ 7 b) 1 ≤ |(x − 1)3 | ≤ 8 − ∗ − ∗ − ∗ − ∗ − ∗ −

More Related Content

Em06 iav

  • 1. ELEMENTARY MATHEMATICS W W L CHEN and X T DUONG c W W L Chen, X T Duong and Macquarie University, 1999. This work is available free, in the hope that it will be useful. Any part of this work may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, with or without permission from the authors. Chapter 6 INEQUALITIES AND ABSOLUTE VALUES 6.1. Some Simple Inequalities Basic inequalities concerning the real numbers are simple, provided that we exercise due care. We begin by studying the effect of addition and multiplication on inequalities. ADDITION AND MULTIPLICATION RULES. Suppose that a, b ∈ R and a < b. Then (a) for every c ∈ R, we have a + c < b + c; (b) for every c ∈ R satisfying c > 0, we have ac < bc; and (c) for every c ∈ R satisfying c < 0, we have ac > bc. In other words, addition by a real number c preserves the inequality. On the other hand, multipli- cation by a real number c preserves the inequality if c > 0 and reverses the inequality if c < 0. Remark. We can deduce some special rules for positive real numbers. Suppose that a, b, c, d ∈ R are all positive. If a < b and c < d, then ac < bd. To see this, note simply that by part (b) above, we have ac < bc and bc < bd. SQUARE AND RECIPROCAL RULES. Suppose that a, b ∈ R and 0 < a < b. Then (a) a2 < b2 ; and (b) a−1 > b−1 . Proof. Part (a) is a special case of our Remark if we take c = a and d = b. To show part (b), note that 1 1 b−a a−1 − b−1 = − = > 0. ♣ a b ab † This chapter was written at Macquarie University in 1999.
  • 2. 6–2 W W L Chen and X T Duong : Elementary Mathematics CAUCHY’S INEQUALITY. For every a, b ∈ R, we have a2 + b2 ≥ 2ab. Furthermore, equality holds precisely when a = b. Proof. Simply note that a2 + b2 − 2ab = a2 − 2ab + b2 = (a − b)2 ≥ 0, and that equality holds precisely when a − b = 0. ♣ We now use some of the above rules to solve inequalities. We shall illustrate the ideas by considering a few examples in some detail. Example 6.1.1. Consider the inequality 4x + 7 < 3. Using the Addition rule and adding −7 to both sides, we obtain 4x < −4. Using one of the Multiplication rules and multiplying both sides by the positive real number 1/4, we obtain x < −1. We have shown that 4x + 7 < 3 =⇒ x < −1. Suppose now that x < −1. Using one of the Multiplication rules and multiplying both sides by the positive real number 4, we obtain 4x < −4. Using the Addition rule and adding 7 to both sides, we obtain 4x + 7 < 3. Combining this with our earlier observation, we have now shown that 4x + 7 < 3 ⇐⇒ x < −1. We can confirm our conclusion by drawing a graph of the line y = 4x + 7 and observing that the part of the line below the horizontal line y = 3 corresponds to x < −1 on the x-axis. y 7 6 y = 4x + 7 5 4 y =3 3 2 1 x -2 -1 1 -1
  • 3. Chapter 6 : Inequalities and Absolute Values 6–3 Example 6.1.2. Consider the inequality −5x + 4 > −1. Using one of the Multiplication rules and multiplying both sides by the negative real number −1, we obtain 5x − 4 < 1. Using the Addition rule and adding 4 to both sides, we obtain 5x < 5. Using one of the Multiplication rules and multiplying both sides by the positive real number 1/5, we obtain x < 1. We have shown that −5x + 4 > −1 =⇒ x < 1. Suppose now that x < 1. Using one of the Multiplication rules and multiplying both sides by the positive real number 5, we obtain 5x < 5. Using the Addition rule and adding −4 to both sides, we obtain 5x − 4 < 1. Using one of the Multiplication rules and multiplying both sides by the negative real number −1, we obtain −5x + 4 > −1. Combining this with our earlier observation, we have now shown that −5x + 4 > −1 ⇐⇒ x < 1. We can confirm our conclusion by drawing a graph of the line y = −5x + 4 and observing that the part of the line above the horizontal line y = −1 corresponds to x < 1 on the x-axis. y 6 5 4 3 y = -5x + 4 2 1 x -1 1 2 -1 y = -1 -2 -3 Example 6.1.3. Consider the inequality x2 ≤ a2 , where a > 0 is fixed. Clearly x = ±a are the only solutions of the equation x2 = a2 . So let us consider the inequality x2 < a2 . Observe first of all that the inequality is satisfied when x = 0. On the other hand, if 0 < x < a, then the Square rule gives x2 < a2 . However, if −a < x < 0, then using one of the Multiplication rules and multiplying all sides by the negative real number −1, we obtain a > −x > 0. It follows from the Square rule that (−x)2 < a2 , so that x2 < a2 . We have now shown that −a ≤ x ≤ a =⇒ x2 ≤ a2 . Suppose now that x > a. Then it follows from the Square rule that x2 > a2 . On the other hand, suppose that x < −a. Using one of the Multiplication rules and multiplying both sides by the negative real number −1, we obtain −x > a. It follows from the Square rule that (−x)2 > a2 , so that x2 > a2 . We have now shown that x < −a or x > a =⇒ x2 > a2 . It now follows that −a ≤ x ≤ a ⇐⇒ x2 ≤ a2 .
  • 4. 6–4 W W L Chen and X T Duong : Elementary Mathematics We can confirm our conclusion by drawing a graph of the parabola y = x2 and observing that the part of the parabola on or below the horizontal line y = a2 corresponds to −a ≤ x ≤ a on the x-axis. y a2 y = x2 x -a a Example 6.1.4. Consider the inequality x2 − 4x + 3 ≤ 0. We can write x2 − 4x + 3 = x2 − 4x + 4 − 1 = (x − 2)2 − 1, so that the inequality is equivalent to (x − 2)2 − 1 ≤ 0, which in turn is equivalent to the inequality (x − 2)2 ≤ 1, in view of the Addition rule. Now write u = x − 2. Then it follows from Example 6.1.3 that −1 ≤ u ≤ 1 ⇐⇒ u2 ≤ 1. Hence −1 ≤ x − 2 ≤ 1 ⇐⇒ (x − 2)2 ≤ 1. Using the addition rule on the inequalities on the left hand side, and using our earlier observation, we conclude that 1≤x≤3 ⇐⇒ x2 − 4x + 3 ≤ 0. We can confirm our conclusion by drawing a graph of the parabola y = x2 − 4x + 3 and observing that the part of the parabola on or below the horizontal line y = 0 corresponds to 1 ≤ x ≤ 3 on the x-axis. y 3 2 y = x2 − 4x + 3 1 x 1 2 3 4 -1 Example 6.1.5. Consider the inequality 1 < 2. x Clearly we cannot have x = 0, as 1/0 is meaningless. We have two cases: (1) Suppose that x > 0. Using one of the Multiplication rules and multiplying both sides by the positive real number x, we obtain the inequality 1 < 2x. Multiplying both sides by the positive real number 1/2, we obtain 1/2 < x. Suppose now that 1/2 < x. Using one of the Multiplication rules
  • 5. Chapter 6 : Inequalities and Absolute Values 6–5 and multiplying both sides by the positive real number 2/x, we obtain the original inequality. We have therefore shown that for x > 0, we have 1 1 x> ⇐⇒ < 2. 2 x (2) Try to use one of the Multiplication rules to show that 1 x<0 ⇐⇒ < 0. x The result is obvious, but the proof is slightly tricky. Combining the two parts, we conclude that 1 1 x<0 or x > ⇐⇒ < 2. 2 x We can confirm our conclusion by drawing a graph of the hyperbola y = 1/x and observing that the part of the hyperbola below the horizontal line y = 2 corresponds to x < 0 together with x > 1/2 on the x-axis. y y = 1/x y=2 2 1 x 1 For the remaining examples in this section, we shall use M to denote an application of one of the Multiplication rules, A to denote an application of the Addition rule and S to denote an application of the Square rule. Example 6.1.6. Consider the inequality x+4 < 3. 2x Clearly we cannot have x = 0. We therefore have two cases: (1) Suppose that x > 0. We have x+4 M A M 4 <3 ⇐⇒ x + 4 < 6x ⇐⇒ 4 < 5x ⇐⇒ < x. 2x 5
  • 6. 6–6 W W L Chen and X T Duong : Elementary Mathematics (2) Suppose that x < 0. We have x+4 M A M 4 <3 ⇐⇒ x + 4 > 6x ⇐⇒ 4 > 5x ⇐⇒ > x. 2x 5 Since the rightmost inequality is always satisfied when x < 0, it follows that the leftmost inequality is always satisfied when x < 0. Combining the two parts, we conclude that 4 x+4 x<0 or x > ⇐⇒ < 3. 5 2x Example 6.1.7. Consider the inequality x2 − 3x + 4 ≤ 1. x+1 Clearly we cannot have x = −1. We therefore have two cases: (1) Suppose that x > −1, so that x + 1 > 0. Then x2 − 3x + 4 M A ≤1 ⇐⇒ x2 − 3x + 4 ≤ x + 1 ⇐⇒ x2 − 4x + 3 ≤ 0. x+1 Recall from Example 6.1.4 that 1≤x≤3 ⇐⇒ x2 − 4x + 3 ≤ 0. (2) Suppose that x < −1, so that x + 1 < 0. Then x2 − 3x + 4 M A ≤1 ⇐⇒ x2 − 3x + 4 ≥ x + 1 ⇐⇒ x2 − 4x + 3 ≥ 0. x+1 It can be deduced from Example 6.1.4 that x≤1 or x ≥ 3 ⇐⇒ x2 − 4x + 3 ≥ 0. Combining the two parts, we conclude that x2 − 3x + 4 x < −1 or 1 ≤ x ≤ 3 ⇐⇒ ≤ 1. x+1 Example 6.1.8. Suppose that a, b ∈ R are non-negative. We shall prove that a+b √ ≥ ab. 2 Suppose on the contrary that this is not true. Then a+b √ S (a + b)2 M < ab =⇒ < ab =⇒ (a + b)2 < 4ab 2 4 A =⇒ (a − b)2 = (a + b)2 − 4ab < 0. But the last inequality is absurd. Example 6.1.9. Suppose that x ∈ R and x > 0. We shall prove that x + x−1 ≥ 2. Suppose on the contrary that this is not true. Then x + x−1 < 2 M A =⇒ x2 + 1 < 2x =⇒ (x − 1)2 = x2 − 2x + 1 < 0. But the last inequality is absurd.
  • 7. Chapter 6 : Inequalities and Absolute Values 6–7 Example 6.1.10. Suppose that x ∈ R. We shall prove that x2 − 4x + 3 ≥ −1. Suppose on the contrary that this is not true. Then A x2 − 4x + 3 < −1 =⇒ (x − 2)2 = x2 − 4x + 4 < 0. But the last inequality is absurd. 6.2. Absolute Values Definition. For every a ∈ R, the absolute value |a| of a is a non-negative real number satisfying a if a ≥ 0; |a| = −a if a < 0. Remark. If we place the number a on the real number line, then the absolute value |a| represents the distance of a from the origin 0. PROPERTIES OF ABSOLUTE VALUES. For every a, b ∈ R, we have (a) |a| ≥ 0; (b) |a| ≥ a; (c) |a|2 = a2 ; (d) |ab| = |a||b|; and (e) |a + b| ≤ |a| + |b|. The graph of the function y = |x| is given below, where a ∈ R is a non-negative real number. y y = |x| a x -a +a The following is easily seen from the graph. RULE CONCERNING REMOVAL OF ABSOLUTE VALUES. Suppose that a ∈ R is non- negative. Then |x| ≤ a if and only if − a ≤ x ≤ +a, and |x| < a if and only if − a < x < +a. Example 6.2.1. The equation |x| = 4 has two solutions x = ±4.
  • 8. 6–8 W W L Chen and X T Duong : Elementary Mathematics Example 6.2.2. The equation |2x + 1| = 5 has two solutions, one satisfying 2x + 1 = 5 and the other satisfying 2x + 1 = −5. Hence x = 2 or x = −3. Example 6.2.3. The inequality |x| < 5 is satisfied precisely when −5 < x < 5. Example 6.2.4. The inequality |2x + 1| ≤ 9 is satisfied precisely when −9 ≤ 2x + 1 ≤ 9; in other words, when −5 ≤ x ≤ 4. √ Example 6.2.5. The equation x2 + 4x + 13 = x − 1 is satisfied only if the right hand side is non- negative, so that we must have x ≥ 1. Squaring both sides, we have x2 + 4x + 13 = (x − 1)2 = x2 − 2x + 1, so that 6x + 12 = 0, giving x = −2. Hence the equation has no real solution x. Example 6.2.6. Consider the inequalities 3 < |x+4| < 7. Note first of all that the inequality |x+4| < 7 holds precisely when −7 < x + 4 < 7; in other words, when −11 < x < 3. On the other hand, the inequality |x + 4| ≤ 3 holds precisely when −3 ≤ x + 4 ≤ 3; in other words, precisely when −7 ≤ x ≤ −1. Hence the inequality |x + 4| > 3 holds precisely when x < −7 or x > −1. It follows that the original inequalities hold precisely when −11 < x < −7 or −1 < x < 3. We can confirm our conclusion by drawing a graph of the function y = |x + 4| and observing that the part of the graph between the horizontal lines y = 3 and y = 7 corresponds to −11 < x < −7 together with −1 < x < 3 on the x-axis. y y = |x + 4| y=7 y=3 x -11 -7 -4 -1 3 Problems for Chapter 6 1. Suppose that α and β are two positive real numbers. The number 1 (α + β) is called the arithmetic √ 2 mean of α and β, while the number αβ is called the geometric mean of α and β. √ a) Prove that αβ ≤ 1 (α + β); in other words, the geometric mean never exceeds the arithmetic 2 mean. b) Show that equality holds in part (a) precisely when α = β. [Remark: The famous arithmetic-mean-geometric-mean theorem states that for any k positive √ integers α1 , . . . , αk , we have k α1 . . . αk ≤ k (α1 + . . . + αk ), and that equality holds precisely when 1 α1 = . . . = αk . The proof is rather complicated when k > 2, and is beyond the scope of our present discussion.] 2. For each of the following inequalities, find all real values of x satisfying the inequality: a) 2x + 4 < 6 b) 5 − 3x > 11 c) 7x + 9 > −5 d) 4x + 4 < 28 e) 2x + 5 < 3 f) 4 − 6x ≥ 10 3. Determine all real values of x for which the inequalities 5 < 2x + 7 ≤ 13 hold.
  • 9. Chapter 6 : Inequalities and Absolute Values 6–9 4. For each of the following inequalities, determine all real values of x for which the inequality holds: a) 6 + x − x2 ≥ 0 b) x2 − 1 > 0 c) x2 − 4 ≤ 0 d) 2 − x − x2 ≥ 0 e) x2 + 2x + 1 ≤ 0 f) (2x + 3)2 ≤ 4 g) (3x − 1)2 > 9 5. For each of the following inequalities, determine all real values of x for which the inequality holds, taking care to distinguish the two cases x > 0 and x < 0, and explain each step of your argument by quoting the relevant rules concerning inequalities: x+4 1 1 a) <3 b) <3 c) −2 < < 3 2x x x 6. For each of the following inequalities, determine all real values of x for which the inequality holds, taking care to distinguish two cases, and explain each step of your argument by quoting the relevant rules concerning inequalities: 2x + 3 4x − 2 4x − 2 a) <1 b) ≥2 c) 2 ≤ <3 3x + 1 x+4 x+4 x2 + 4x − 7 7. Show that ≥ 2 precisely when −5 ≤ x < −4 or x ≥ 3. x+4 8. For each of the following inequalities, determine all real values of x for which the inequality holds: x2 − 5x + 3 x2 − 4x + 1 a) ≤1 b) ≤1 x−2 x−3 9. Find all solutions of the inequality |x + 2| < 6, and confirm your answer by drawing a suitable picture. 10. For each of the following inequalities, determine all real values of x for which the inequality holds: a) 1 < |3x − 5| ≤ 7 b) 1 ≤ |(x − 1)3 | ≤ 8 − ∗ − ∗ − ∗ − ∗ − ∗ −