REVIEW OF FUNDAMENTALS
R.R Review Exercises for Chapter
SOLUTIONS TO EVERY OTHER ODD EXERCISE
1. Subtract 2 from both sides t o get 3x = -2 . Then divide by 3 to
get x = -2/3 .
5. Expansion gives (x + 1)2 - (x - 1)2 = 2 = (x2 + 2x + 1) - (x2 - 2x + 1) =
4x . Divide both sides by 4 to get x = 1/2 .
9. 8x + 2 > 0 is equivalent is 8x > -2 . Divide both sides by 8 to get
x > -1/4 .
13. Expand to get . Add 1
to both sides to get 2x > 3 and divide by 2 to get x > 3/2 .
17. x2 < 1 is equivalent to x2 - 1 < 0 or (x + l ) ( x - 1) < 0 . One possible
solution is x < -1 and x > 1 , but this is not possible. Another
possibility is x > -1 and x < 1 . Thus, the solution is -1 < x < 1 ,
i . e . , x ∈ (-1,1) .
21. means , i . e . , or .
For , we get . For , we get
. Therefore, the solution is or ,
25. x3 ∈ (-8,27) implies x ∈ (-2,3) , so x < 10 and x ∈ (-2,3) means
-2 < x < 3 ; i . e . , x ∈(-2,3) .
29. is equivalent t o or . Dividing
by -2 reverses the inequality and yields . Also, 3x - 22 > 0
is equivalent t o 3x > 22 or x > 22/3 . Thus, the solution is
45. The distance between P1 and P2 is
In this case , it is .
49. The point-point form of a line is .
In t his case, the line is y = -1 + [ ( 3 - (-1))/(7 - 1 / 2 ) ] ( x - 1/2) =
53. The point-slope form of a line is . In t his case, the
line is y = 13 + (-3)(x - 3/4) = -3x + 61/4 or 4y + 3x = 61 .
57. The slope of 5y + 8x = 3 is -8/5 . Thus, the slope of a perpendicular
line is 5/8 , and the line passing through ( 1 ,1) is y = 1 + (5/8) (x - 1) ,
using point-slope form of the line . The line is y = 5x/8 + 3/8 or
61. The equation of the circle with center (a,b) and radius r is
. In this case, the circle is
(y - 5)2 = 82 or (x2 - 24x + 144) + (y2 - 10y + 25) = 64 or
x2 - 24x + y2 - 10y + 105 = 0 .
Complete the square to get
. This is a parabola
opening upward with vertex at (-2/3,2/3) .
69. Solve the equations simultaneously. Substitute y = x in to x2 + y2 = 4
to get 2x2 = 4 , i . e . , x2 = 2 or . Thus, the points of inter -
section are and
This is the graph of y = 3x if ;
it is y = -3x if x < 0 .
In addition t o the points in
part (a), we have f(-1.5) =
-0.1875 ; f(-1) = 0 ; f(-0.5) =
0.0375 ; f(0.5) = -0.0375 ;
81. If a vertical line passes through two
points of the graph, it is not a
function. Thus, (a) and (c) a r e functions.
TEST FOR CHAPTER R
1. True or false :
(a) I f a > b > 0 , then 1 / a > 1 / b .
(b) The interval (2,4) contains 3 integers.
(c) If a > b > 0 , then ac > bc for constant c .
(d) The domain of is x≥ 0 .
(e) The line x = 2 has slope zero.
2.Express the solution set of x2 + 3x + 2 ≥ 0 in terms of
3.Write equations for the following lines:
(a) The line going through (1,3) and (2,4) .
(b) The line with slope 5 and passing through (-3,2) .
(c) The line with slope -1 and y-intercept 1/2 .
4.Sketch the graph of .
5.Do the following equations describe a circle or a parabola?
6.(a) Complete the square for the equation y2 + 4y + 3 = 0 .
(b) Solve the equation in part (a) .
7.Sketch the graph of y = Ix2 - 1 I .
8.(a) Find the points of intersection of the graphs of y = x2 and
(b) Find the distance between the intersection points.
9.Factor the following expressions.
10.Dumb Donald had heard how wonderful chocolate mousse tasted, so he
decided to go on a hunting trip at Moose Valley. When he finally
spotted a moose, he chased the moose in circles around a tree located
at (-4,-2). Unfortunately for Dumb Donald, the moose ran much faster,
caught up with Dumb Donald, and trampled him. Their path was radius 7
from the tree. What is the equation of the circle?
ANSWERS TO CHAPTER TEST
1. (a) False; 1/a < 1/b .
(b) False; 2 and 4 are excluded from the interval.
(c) False; c must be positive for ac > bc .
(e) False; vertical lines do not have slopes.
5. (a) Circle