4.2: Inequalities and modulus

The transition from school to university mathematics is in many ways marked by a shift from simple variables, equations and functions, to conditions and analysis involving inequalities and modulus.

Problem 100 What is | − x| equal to: x or −x? (What if x is negative?)

4.2.1 Geometrical interpretation of modulus, of inequalities, and of modulus inequalities

Problem 101

(a) Mark on the coordinate line all those points x in the interval [0,1) which have the digit “1” immediately after the decimal point in their decimal expansion. What fraction of the interval [0,1) have you marked?

Note: “[0,1)” denotes the set of all points between 0 and 1, together with 0, but not including 1. [0,1] denotes the interval including both endpoints; and (0,1) denotes the interval excluding both endpoints.

(b) Mark on the interval [0,1) all those points x which have the digit “1” in at least one decimal place. What fraction of the interval [0,1) have you marked?

(c) Mark on the interval [0,1) all those points x which have a digit “1” in at least one position of their base 2 expansion. What fraction of the interval [0,1) have you marked?

(d) Mark on the interval [0,1) all those points x which have a digit “1” in at least one position of their base 3 expansion. What fraction of the interval [0,1) have you marked?

Problem 102 Mark on the coordinate line all those points x for which two of the following inequalities are true, and five are false:

$$x>1,x>2,x>3,x>4,x>5,x>6,x>7.$$

Problem 103 Mark on the coordinate line all those points x for which

$$|x-5|=3.$$

Problem 104

(a) Mark on the coordinate line all those points x for which two of the following inequalities are true, and five are false:

$$\left|x\right|>1,\left|x\right|>2,\left|x\right|>3,\left|x\right|>4,\left|x\right|>5,\left|x\right|>6,\left|x\right|>7.$$

(b) Mark on the coordinate line all those points x for which two of the following inequalities are true, and five are false:

$$|x-1|>1,|x-2|>2,|x-3|>3,|x-4|>4,|x-5|>5,|x-6|>6,|x-7|>7.$$

Problem 105 Mark on the coordinate line all those points x for which

$$|x+1|+|x+2|=2.$$

Problem 106 Find numbers a and b with the property that the set of solutions of the inequality

$$|x-a|<b$$

consists of the interval (−1, 2).

Problem 107

(a) Mark on the coordinate plane all points (x, y) satisfying the inequality

$$|x-y|<3.$$

(b) Mark on the coordinate plane all points (x, y) satisfying the inequality

$$|x-y+5|<3.$$

(c) Mark on the coordinate plane all points (x, y) satisfying the inequality

$$|x-y|<|x+y|.$$

4.2.2 Inequalities

Problem 108 Suppose real numbers a, b, c, d satisfy $\frac{a}{b}<\frac{c}{d}$ .

(i) Prove that

$$\frac{a}{b}<\frac{\left(\frac{a}{b}+\frac{c}{d}\right)}{2}<\frac{c}{d}.$$

(ii) If b, d > 0, prove that

$$\frac{a}{b}<\frac{a+c}{b+d}<\frac{c}{d}.$$

Problem 109 (Farey series) When the fully cancelled fractions in [0,1] with denominator ≤ n are arranged in increasing order, the result is called the Farey series (or Farey sequence) of order n.

Order 1: $\frac{0}{1}<\frac{1}{1}$

Order 2: $\frac{0}{1}<\frac{1}{2}<\frac{1}{1}$

Order 3: $\frac{0}{1}<\frac{1}{3}<\frac{1}{2}<\frac{2}{3}<\frac{1}{1}$

Order 4: $\frac{0}{1}<\frac{1}{4}<\frac{1}{3}<\frac{1}{2}<\frac{2}{3}<\frac{3}{4}<\frac{1}{1}$

(a) Write down the full Farey series (or sequence) of order 7.

(b) (i) Imagine the points 0.1, 0.2,0.3,..., 0.9 dividing the interval [0,1] into ten subintervals of length $\frac{1}{10}$ . Now insert the eight points corresponding to

$$\frac{1}{9},\frac{2}{9},\frac{3}{9},\dots ,\frac{8}{9}.$$

Into which of the ten subintervals do they fall?

(ii) Imagine the n points

$$\frac{1}{n+1},\frac{2}{n+1},\frac{3}{n+1},\dots ,\frac{n}{n+1}$$

dividing the interval [0,1] into n + 1 subintervals of length $\frac{1}{n+1}$ . Now insert the n − 1 points

$$\frac{1}{n},\frac{2}{n},\frac{3}{n},\dots ,\frac{n-1}{n}.$$

Into which of the n + 1 subintervals do they fall?

(iii) In passing from the Farey series of order n to the Farey series of order n + 1, we insert fractions of the form $\frac{k}{n+1}$ between certain pairs of adjacent fractions in the Farey series of order n. If $\frac{a}{b}<\frac{c}{d}$ are adjacent fractions in the Farey series of order n, prove that, when adding fractions for the Farey series of order n + 1, at most one fraction is inserted between $\frac{a}{b}\mathrm{a}\mathrm{n}\mathrm{d}\frac{c}{d}$ .

(c) Note: It is worth struggling to prove the two results in part (c). But do not be surprised if they prove to be elusive – in which case, be prepared to simply use the result in part (c)(ii) to solve part (d).

(i) In the Farey series of order n the first two fractions are $\frac{0}{1}<\frac{1}{n}$ , and the last two fractions are $\frac{n-1}{n}<\frac{1}{1}$ . Prove that every other adjacent pair of fractions $\frac{a}{b}<\frac{c}{d}$ in the Farey series of order n satisfies bd > n.

(ii) Let $\frac{a}{b}<\frac{c}{d}$ be adjacent fractions in the Farey series of order n. Prove (by induction on n) that bc − ad = 1.

(d) Prove that if

$$\frac{a}{b}<\frac{c}{d}<\frac{e}{f}$$

are three successive terms in any Farey series, then

$$\frac{c}{d}=\frac{a+e}{b+f}.$$

Problem 110 Solve the following inequalities.

(a) $x+\frac{1}{x}<2$

(b) $x⩽1+\frac{2}{x}$

(c) $\sqrt{x}<x+\frac{1}{4}$

Problem 111

(a) The sum of two positive numbers equals 5. Can their product be equal to 7?

(b) (Arithmetic mean, Geometric mean, Harmonic mean, Quadratic mean) Prove that, if a,b > 0, then

$$\begin{array}{l}\frac{2}{\left[\frac{1}{a}+\frac{1}{b}\right]}=\frac{2ab}{a+b}⩽\sqrt{ab}⩽\frac{a+b}{2}⩽\sqrt{\frac{a^2+b^2}{2}}\\ (\text{HM}⩽\text{GM}⩽\text{AM}⩽\text{QM})\end{array}$$

Problem 112 The two hundred numbers

$$1,2,3,4,5,\dots ,200$$

are written on the board. Students take turns to replace two numbers a, b from the current list by their sum divided by $\sqrt{2}$ . Eventually one number is left on the board. Prove that the final number must be less than 2000.