294 MCGRAW-HILL’S SAT
Concept Review 6
- < 2. > 3. < 4. >
- Whenever xis negative.
- You multiply or divide by a negative on both sides.
- Whenever yis greater than x(regardless of sign).
- If the number of negatives in the term is odd, then
the result is negative. If the number of negatives
in the term is even, then the result is positive. Re-
member to add the exponents of all the negative
numbers in the term.
- Notice that y−xis the opposite of x−y.So
(x−y)(x−y) is the opposite of (x−y)(y−x).
- Notice that y−xis the opposite of x−y.So
10.−1. Notice that 9x−18 and 18 − 9 xmust be oppo-
sites, and the quotient of non-zero opposites is
always −1. (−5/5 =−1, 20/− 20 =−1, etc.)
11.− 9 y^2. − 13 y^2 −(− 4 y^2 )
Change to addition: − 13 y^2 + 4 y^2
Commute: 4 y^2 +− 13 y^2
Change to subtraction: 4 y^2 − 13 y^2
Swap and negate: −(13y^2 − 4 y^2 ) =− 9 y^2
12.−16/15.
Simplify fractions:Factor out −1:Zip-zap-zup:13.−1/16. Since there are an odd number (19)
of negatives, the result is negative. Notice, too,
that the powers of 5 cancel out.
14.− 6 15. 35x^17
16.a> bonly if x< 2 (and so 2 −xis positive).
a< bonly if x> 2 (and so 2 −xis negative).
17.x< 1 18. x≥ 9
19.x> − 9 20. x< − 1−
+
=−
10 6
15
16
15
−+
⎛
⎝⎜
⎞
⎠⎟
2
3
2
5
−−
2
3
2
5
2
3
2
− 5
−
−
−
Answer Key 6: Negatives
SAT Practice 6
- B −(−b−b−b−b)
Convert to addition: −(−b+−b+−b+−b)
Distribute −1: (b+b+b+b)
Simplify: 4 b
2.D Notice that m−1 > m−2 > m−3. If the prod-
uct is positive, then all three terms must be posi-
tive or one must be positive and the other two
negative. They would all be positive only if m> 3,
but no choice fits. If two terms are negative and
one positive, then, by checking, 1 < m< 2.
3.B Don’t forget the order of operations: powers
before subtraction! −w^2 −(−w)^2
Simplify power: −w^2 −w^2
Change to addition: −w^2 +−w^2
Simplify: − 2 w^2
4.B If m/nis positive, then mand nmust have the
same sign. Using m=−2 and n=−1 disproves
statements I and III. Statement II must be true
because mand nmust have the same sign.
5.C The example x=−1 and y=1 disproves state-
ment I. Substituting −yfor x(because an expres-
sion can always be substituted for its equal) and
simplifying in statements II and III proves that
both are true.
6.C Since mis negative, n^4 p^5 must be negative be-
cause the whole product is positive. ncan be
either positive or negative; n^4 will be positive in
either case. Therefore, p^5 must be negative.7.D Since x−yis always the opposite of y−x,
(a−b)(c−d)(e−f)(x) =(b−a)(d−c)(f−e).
Substitute: =−(a−b) ×−(c−d) ×−(e−f)
Simplify: =−(a−b)(c−d)(e−f)
By comparing the two sides, xmust equal −1.
8.E If the two sums are equal, then the sum of the
integers from nto 14, which are not included in
the first sum, must “cancel out.” That can only
happen if nis −14.
9.C Simplify the first inequality by dividing both
sides by −2. (Don’t forget to “flip” the inequality!)
This gives x> 3.5. The example of x=4 and y=− 1
disproves statement I. Since xmust be greater
than y, statement II must be true. Since xis
greater than 3.5, it must certainly be greater
than3, so statement III must be true.10.D The sequence follows the pattern (−1, 1, 1, −1),
(−1, 1, 1, −1), (−1, 1, 1, −1),.... Since the pattern
is four terms long, it repeats 57 ÷ 4 =14 times,
with a remainder of 1 (the remainder shows that
it includes the first term of the 15th repetition),
which means it includes 14(2) + 1 =29 negatives.