Letaandbbe any two numbers on the real line. Then we call the set of all numbers that lie betweenaandbaninterval. In many situations, it is important to distinguish between the intervals that include their endpoints and the intervals that do not. Whena < b, there are four different intervals that all haveaandbas endpoints, as shown in Table 1.
Table 1
Notation Name The interval consists
of allxsatisfying:
(a, b) Theopeninterval fromatob. a < x < b [a, b] Theclosedinterval fromatob. a≤x≤b (a, b] Ahalf-openinterval fromatob. a < x≤b [a, b) Ahalf-openinterval fromatob. a≤x < b
Note that an open interval includes neither of its endpoints, but a closed interval includes both of its endpoints. A half-open interval contains one of its endpoints, but not both. All four intervals, however, have the same length,b−a.
We usually illustrate intervals on the number line as in Fig. 1, with included endpoints represented by dots, and excluded endpoints at the tips of arrows.
⫺2
⫺3
⫺4
⫺5 ⫺1 0 1 2 3 4 5 6 7
A B C
Figure 1 A=[−4,−2],B=[0,1), andC=(2,5)
The intervals mentioned so far are allbounded intervals. We also use the word “interval” to signify certain unbounded sets of numbers. For example, we have
[a,∞)=all numbersxwithx ≥a (−∞, b)=all numbersxwithx < b
with “∞” as the common symbol for infinity. The symbol∞is not a number at all, and therefore the usual rules of arithmetic do not apply to it. In the notation [a,∞), the symbol
∞is only intended to indicate that we are considering the collection ofallnumbers larger than or equal toa, without any upper bound on the size of the number. Similarly,(−∞, b) has no lower bound. From the preceding, it should be apparent what we mean by(a,∞) and(−∞, b]. The collection of all real numbers is also denoted by the symbol(−∞,∞).
Absolute Value
Letabe a real number and imagine its position on the real line. The distance betweena and 0 is called theabsolute valueofa. Ifa is positive or 0, then the absolute value is the numberaitself; ifais negative, then because distance must be positive, the absolute value is equal to the positive number−a.
Theabsolute valueofais denoted by|a|, and
|a| =
a ifa ≥0
−a ifa <0 (1)
For example,|13| =13,|−5| = −(−5)=5,|−1/2| =1/2, and|0| =0. Note in particular that|−a| = |a|.
NOTE 1 It is a common fallacy to assume thatamust denote a positive number, even if this is not explicitly stated. Similarly, on seeing−a, many students are led to believe that this expression is always negative. Observe, however, that the number−ais positive when a itself is negative. For example, ifa = −5, then−a = −(−5)=5. Nevertheless, it is often a useful convention in economics to define variables so that, as far as possible, their values are positive rather than negative.
E X A M P L E 1
(a) Compute|x−2|forx = −3,x =0, andx=4.
(b) Rewrite|x−2|using the definition of absolute value.
Solution:
(a) Forx = −3,|x−2| = |−3−2| = |−5| =5. Forx=0,|x−2| = |0−2| = |−2| =2.
Forx =4,|x−2| = |4−2| = |2| =2.
(b) According to the definition (1),|x−2| =x−2 ifx−2≥0, that is,x ≥2. However,
|x−2| = −(x−2)=2−xifx−2<0, that is,x <2. Hence,
|x−2| =
x−2, ifx ≥2 2−x, ifx <2
Letx1andx2be two arbitrary numbers. Thedistancebetweenx1andx2on the number line isx1−x2ifx1≥x2, and−(x1−x2)ifx1 < x2. Therefore, we have
|x1−x2| = |x2−x1| =distancebetweenx1andx2on the number line (2)
S E C T I O N 1 . 7 / I N T E R V A L S A N D A B S O L U T E V A L U E S 31 In Fig. 2 we have indicated geometrically that the distance between 7 and 2 is 5, whereas the distance between−3 and−5 is equal to 2, because|−3−(−5)| = |−3+5| = |2| =2.
−7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7
|−3 − (−5)| = 2 |7 − 2| = 5
Figure 2 The distances between 7 and 2 and between−3 and−5.
Suppose|x| =5. What values canxhave? There are only two possibilities: eitherx =5 orx = −5, because no other numbers have absolute values equal to 5. Generally, ifa is greater than or equal to 0, then|x| =ameans thatx =aorx= −a. Because|x| ≥0 for allx, the equation|x| =ahas no solution whena <0.
If a is a positive number and |x| < a, then the distance fromx to 0 is less thana.
Furthermore, whena is nonnegative, and|x| ≤a, the distance fromx to 0 is less than or equal toa. In symbols:
|x|< a means that −a < x < a (3)
|x| ≤a means that −a≤x≤a (4)
E X A M P L E 2 Find all thexsuch that
|3x−2| ≤5
Check first to see if this inequality holds forx= −3,x=0,x =7/3, andx =10.
Solution: Forx = −3,|3x−2| = |−9−2| =11; forx =0,|3x−2| = |−2| =2; for x =7/3,|3x−2| = |7−2| =5; and forx =10,|3x−2| = |30−2| =28. Hence, the given inequality is satisfied forx =0 andx =7/3, but not forx = −3 andx=10.
From (4) the inequality|3x−2| ≤5 means that−5≤3x−2≤5. Adding 2 to all three expressions gives
−5+2≤3x−2+2≤5+2 or−3≤3x ≤7. Dividing by 3 gives−1≤x≤7/3.
P R O B L E M S F O R S E C T I O N 1 . 7
1. Calculate|2x−3|forx=0, 1/2, and 7/2.
2. (a) Calculate|5−3x|forx= −1,2, and 4.
(b) Solve the equation|5−3x| =0.
(c) Rewrite|5−3x|by using the definition of absolute value.
3. Determinexsuch that
(a) |3−2x| =5 (b) |x| ≤2 (c) |x−2| ≤1 (d) |3−8x| ≤5 (e) |x|>√
2 (f) |x2−2| ≤1
4. A 5-metre iron bar is to be produced. The bar may not deviate by more than 1 mm from its stated length. Write a specification for the bar’s lengthxin metres: (a) by using a double inequality;
(b) with the aid of an absolute-value sign.
R E V I E W P R O B L E M S F O R C H A P T E R 1
1. (a) What is three times the difference between 50 andx?
(b) What is the quotient betweenxand the sum ofyand 100?
(c) If the price of an item isaincluding 20% VAT (value added tax), what is the price before VAT?
(d) A person buysx1,x2, andx3units of three goods whose prices per unit are respectively p1, p2, andp3. What is the total expenditure?
(e) A rental car costsFdollars per day in fixed charges andbdollars per kilometre. How much must a customer pay to drivexkilometres in 1 day?
(f) A company has fixed costs ofF dollars per year and variable costs ofcdollars per unit produced. Find an expression for the total cost per unit (total average cost) incurred by the company if it producesxunits in one year.
(g) A person has an annual salary of $Land then receives a raise ofp% followed by a further increase ofq%. What is the person’s new yearly salary?
2. Express as single real numbers in decimal notation:
(a) 53 (b) 10−3 (c) 1
3−3 (d) −1
10−3 (e) 3−233 (f) (3−2)−3 (g) −
5 3
0
(h)
−1 2
−3
3. Which of the following expressions are defined, and what are their values?
(a) (0+2)0 (b) 0−2 (c) (10)0
(0+1)0 (d) (0+1)0 (0+2)0 4. Simplify:
(a) (232−5)3 (b) 2
3
−1
− 4
3
−1
(c) (3−2−5−1)−1 (d) (1.12)−3(1.12)3
⊂SM⊃5. Simplify:
(a) (2x)4 (b) (2−1−4−1)−1 (c) 24x3y2z3 4x2yz2 (d)
−(−ab3)−3(a6b6)23
(e) a5ãa3ãa−2
a−3ãa6 (f) x
2
3
ã 8 x−2
−3
6. Compute: (a) 12% of 300 (b) 5% of 2000 (c) 6.5% of 1500
R E V I E W P R O B L E M S F O R C H A P T E R 1 33 7. Give economic interpretations to each of the following expressions and then use a calculator to
find the approximate values:
(a) 100ã(1.01)8 (b) 50 000ã(1.15)10 (c) 6000ã(1.03)−8
8. (a) $100 000 is deposited into an account earning 8% interest per year. What is the amount after 10 years?
(b) If the interest rate is 8% each year, how much money should you have deposited in a bank 6 years ago to have $25 000 today?
⊂SM⊃9. Expand and simplify:
(a) a(a−1) (b) (x−3)(x+7) (c) −√ 3√
3−√ 6
(d) 1−√
22
(e) (x−1)3 (f) (1−b2)(1+b2) (g) (1+x+x2+x3)(1−x) (h) (1+x)4 10. Complete the following:
(a) x−1y−1=3 impliesx3y3= ã ã ã (b) x7=2 implies(x−3)6(x2)2= ã ã ã (c)
xy z
−2
=3 implies z
xy
6
= ã ã ã (d) a−1b−1c−1=1/4 implies(abc)4= ã ã ã 11. Factor the expressions
(a) 25x−5 (b) 3x2−x3y (c) 50−x2 (d) a3−4a2b+4ab2
⊂SM⊃12. Factor the expressions
(a) 5(x+2y)+a(x+2y) (b) (a+b)c−d(a+b) (c) ax+ay+2x+2y (d) 2x2−5yz+10xz−xy (e) p2−q2+p−q (f) u3+v3−u2v−v2u 13. Compute the following without using a calculator:
(a) 161/4 (b) 243−1/5 (c) 51/7ã56/7 (d) (48)−3/16 (e) 641/3+√3
125 (f) (−8/27)2/3 (g) (−1/8)−2/3+(1/27)−2/3 (h) 1000−2/3
√3
5−3 14. Solve the following equations forx:
(a) 22x=8 (b) 33x+1=1/81 (c) 10x2−2x+2=100 15. Find the unknownxin each of the following equations:
(a) 255ã25x=253 (b) 3x−3x−2=24 (c) 3xã3x−1=81 (d) 35+35+35=3x (e) 4−6+4−6+4−6+4−6=4x (f) 226−223
226+223 =x 9
⊂SM⊃16. Simplify: (a) s
2s−1− s
2s+1 (b) x
3−x −1−x x+3− 24
x2−9 (c) 1 x2y− 1
xy2 1 x2− 1
y2
⊂SM⊃17. Reduce the following fractions:
(a) 25a3b2
125ab (b) x2−y2
x+y (c) 4a2−12ab+9b2
4a2−9b2 (d) 4x−x3 4−4x+x2 18. Solve the following inequalities:
(a) 2(x−4) <5 (b) 1
3(y−3)+4≥2 (c) 8−0.2x≤4−0.1x 0.5 (d) x−1
−3 >−3x+8
−5 (e) |5−3x| ≤8 (f) |x2−4| ≤2 19. Using a mobile phone costs $30 per month, and an additional $0.16 per minute of use.
(a) What is the cost for one month if the phone is used for a total ofxminutes?
(b) What are the smallest and largest numbers ofhoursyou can use the phone in a month if the monthly telephone bill is to be between $102 and $126?
20. If a rope could be wrapped around the Earth’s surface at the equator, it would be approximately circular and about 40 million metres long. Suppose we wanted to extend the rope to make it 1 metre above the equator at every point. How many more metres of rope would be needed? (The circumference of a circle with radiusris 2π r.)
21. (a) Prove thata+aãp 100 −
a+aãp 100
ãp
100 =a
1− p
100 2
.
(b) An item initially costs $2000 and then its price is increased by 5%. Afterwards the price is lowered by 5%. What is the final price?
(c) An item initially costsadollars and then its price is increased byp%. Afterwards the (new) price is lowered byp%. What is the final price of the item? (After considering this problem, look at the expression in part (a).)
(d) What is the result if one firstlowersa price byp% and thenincreasesit byp%?
22. (a) Ifa > b, is it necessarily true thata2> b2? (b) Show that ifa+b >0, thena > bimpliesa2> b2.
23. (a) Ifa > b, use numerical examples to check whether 1/a >1/b, or 1/a <1/b.
(b) Prove that ifa > bandab >0, then 1/b >1/a.
24. Prove that (i)|ab| = |a| ã |b|and (ii)|a+b| ≤ |a| + |b|, for all real numbersaandb. (The inequality in (ii) is called thetriangle inequality.)
⊂SM⊃25. Consider an equilateral triangle, and letP be an arbitrary point within the triangle. Leth1,h2, andh3be the shortest distances fromPto each of the three sides. Show that the sumh1+h2+h3 is independent of where pointPis placed in the triangle. (Hint:Compute the area of the triangle as the sum of three triangles.)
2 I N T R O D U C T O R Y T O P I C S I I :
E Q U A T I O N S
. . .and mathematics is nourished by dreamers
—as it nourishes them.
—D’Arcy W. Thompson (1940)
Science uses mathematical models, which often include one or more equations whose solu- tion determines the magnitudes of some variables we would like to understand better.
Economics is no exception. Accordingly, this chapter considers some types of equation that appear frequently in economic models.
Many students are used to dealing with algebraic expressions and equations involving only onevariable (usuallyx). Often, however, they have difficulties at first in dealing with expressions involving several variables with a wide variety of names, and denoted by different letters. For economists, however, it is very important to be able to handle such algebraic expressions and equations with ease.