Sat math Essentials

4.
d.
If
b
3
= –64, then, taking the cube root of both
sides,
b
= –4. Substitute –4 for 
b
in the second
equation:
b
2
– 3
b
– 4 = (–4)
2
– 3(–4) – 4 = 16 +
12 – 4 = 24.
5.
e.
The point that represents a number of eggs
found that is greater than the number of min-
utes that has elapsed is the point that has a 
y
value that is greater than its 
x
value. Only point
E
lies farther from the horizontal axis than it lies
from the vertical axis. At point 
E
, more eggs
have been found than the number of minutes
that has elapsed.
6.
c.
The midpoint of a line is equal to the average
of the 
x
-coordinates and the average of the 
y
-coordinates of the endpoints of the line. The
midpoint of the line with endpoints at (6,0) and
(6,–6) is (
6 +
2
6
,
0 +
2
–6
) = (
1
2
2
,–
6
2
) = (6,–3).
7.
a.
The number of yellow marbles, 24, is 
2
8
4
= 3
times larger than the number of marbles given
in the ratio. Multiply each number in the ratio
by 3 to find the number of each color of
marbles. There are 3(3) = 9 red marbles and
4(3) = 12 blue marbles. The total number of
marbles in the sack is 24 + 9 + 12 = 45.
8.
a.
The equation 
y
= 
x
2
x
–
2
9
–
x
3
–
6
36
is undefined when
its denominator,
x
2
– 9
x
– 36, evaluates to zero.
The 
x
values that make the denominator evalu-
ate to zero are not in the domain of the equa-
tion. Factor 
x
2
– 9
x
– 36 and set the factors equal
to zero:
x
2
– 9
x
– 36 = (
x
– 12)(
x
+ 3);
x
– 12 =
0,
x
= 12;
x
+ 3 = 0,
x
= –3.