By Knut Sydsæter, Peter Hammond, Arne Strøm

This article presents a useful creation to the mathematical instruments that undergraduate economists desire. The assurance is accomplished, starting from easy algebra to extra complicated fabric, when targeting the entire middle subject matters which are frequently taught in undergraduate classes on arithmetic for economists.

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**Example text**

For Econ. 6 / INEQUALITIES 27 WARNING 2 It is vital that you really understand the method of sign diagrams. A common error is illustrated by the following example. Find the solution set for (x − 2) + 3(x + 1) ≤0 x+3 “Solution”: We construct the sign diagram: −3 −2 −1 x−2 1 2 ◦ 3(x + 1) x+3 (x − 2) + 3(x + 1) x+3 0 ◦ ◦ ∗ ◦ ◦ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ Wrong! According to this diagram, the inequality should be satisﬁed for x < −3 and for −1 ≤ x ≤ 2. However, for x = −4 (< −3), the fraction reduces to 15, which is positive.

Because (−1)x = −x, −(a + b − c + d) = −a − b + c − d In words: When removing a pair of parentheses with a minus in front, change the signs of all the terms within the parentheses—do not leave any out. We saw how to multiply two factors, (a +b) and (c +d). How do we compute such products when there are several factors? Here is an example: (a + b)(c + d)(e + f ) = (a + b)(c + d) (e + f ) = ac + ad + bc + bd (e + f ) = (ac + ad + bc + bd)e + (ac + ad + bc + bd)f = ace + ade + bce + bde + acf + adf + bcf + bdf Alternatively, write (a + b)(c + d)(e + f ) = (a + b) (c + d)(e + f ) , then expand and show that you get the same answer.

You probably know the meaning of a x if x = 1/2. In fact, if a ≥ 0 and x = 1/2, then √ √ we deﬁne a x = a 1/2 as equal to a, the square root of a. Thus, a 1/2 = a is deﬁned as the nonnegative number that when multiplied by itself gives a. This deﬁnition makes sense because a 1/2 · a 1/2 = a 1/2+1/2 = a 1 = a. Note that a real number multiplied by itself must always be ≥ 0, whether that number is positive, negative, or zero. Hence, a 1/2 = √ a (valid if a ≥ 0) √ 1 = 15 because For example, 16 = 161/2 = 4 because 42 = 16 and 25 If a and b are nonnegative numbers (with b = 0 in (ii)), then (i) ab = a b (ii) 1 5 · 1 5 = 1 25 .