SOLUTION SET:

MATH 2 – INTERMEDIATE ALGEBRA

1. A (1,2) B (9,2) C (1,k) Segment AB = AC AB = X2 – X1 = 9-1 = 8 AC = Y2 – Y1 = k – 2 = 8 k – 2 = 8; k = 8 + 2 k = 10 answer: B

2. line parallel to: y = 2x – 5

Passing through pt. (1, 1) Since y = mx + b; m = 2

y = 2x + b; then substitute the coordinates of pt. (1, 1) 1 = 2(1) + b; b = 1 – 2

b = -1 y = 2x - 1 answer: B

3. Slope of the line containing pts. (2, -4) and (-5, 7)

Slope = m =
1
2
1
2
*x*
*x*
*y*
*y*
=
2
5
)
4
(
7
=
-7
11
answer: B

4. inspect the graph: the line passes through pts. (1, 2) and (4, -4)

slope = m = 1 4 2 4 = 3 6 = -2

y = -2x + b; then substitute any point. We use pt. (1, 2)
2 = -2(1) +b; b = 2 + 2
b = 4
y = -2x + 4 or y = 4 – 2x
answer: A
5. a = b + ½ =
2
3
*b*
b + ½ =
2
3
*b*

; multiply both sides by 2 2b + 1 = b + 3; 2b – b = 3 -1

b = 2

a = b + ½ = 2 + ½ = 2 ½ = 5/2 answer: C

2x + 4y – 5(3) = 19

2x + 4y = 19 + 15; 2x + 4y = 34, divide both sides by 2 x + 2y = 17, thus, x + 2y + z = 17 + 3 = 20

answer: A
7. 32/n _{= }*n* _{3}2 = *n* _{9}

answer: B

8. 21*q*_{ = 3 + } *q*_{ ; we raise both sides to the second power}

21 + q = (3 + *q*_{)}2_{ = 9 + 6} _{q}_{ + q}
21 – 9 + q – q = 6 *q*
12 = 6 *q*_{, } *q*_{ = 2}
q = 4
answer: B
9.
3
1
*a* + 2
3
*a*
*a*
=
2
3
1
*a*
*a*
, then we multiply by
2
2
*a*
*a*
2
3
1
*a*
*a*
x
2
2
*a*
*a*
=

##

##

##

2 2 3 1 *a*

*a*

*a*answer: D 10. 2 2

*x*- 4 3 2

_{}

*x*= 2 2

*x*-

###

2###

2###

3 *x*

*x*=

###

2###

2###

3 ) 2 ( 2 *x*

*x*

*x*) 2 )( 2 ( 3 4 2

*x*

*x*

*x*=

_{(}

_{x}_{}2

_{2}

*x*

_{)(}

*7*

_{x}_{}

_{2}

_{)}= 4 7 2 2

_{}

*x*

*x*answer: D 11.

_{ab – 1 – b + a = ( ab – b) + ( a – 1)}b ( a – 1) + ( a – 1) = ( b + 1) ( a – 1) answer: C 12.

_{x}2

_{ – 6x + 5 = 0}( x – 5)( x – 1) = 0 A. x2

_{ + 1 = 0}B. x2

_{ – x – 2 = 0, ( x -2 )( x + 1) = 0}C. 2x2

_{ – 2 = 0, 2( x}2

_{ – 1 ) = 0, 2( x + 1)( x – 1) = 0}D. x2

_{ – 2x – 3 = 0, ( x – 3 )( x + 1) = 0}answer: C

13. _{perfect square trinomial: a}2_{x}2 _{ 2abx + b}2_{ = ( ax } _{ b )}2

4x2_{ – 20x + 25 = ( 2x – 5 )}2

answer: B

x2_{ + 2xy + y}2_{ = 20}

x2_{ + 2(4) + y}2_{ = 20}

x2_{ + y}2_{ = 20 – 8 = 12}

answer: C

15. _{-6 is a solution to x}2_{ + 5x + k = 7, substitute -6 to all values of x}

(-6)2_{ + 5(-6) + k = 7, k = 7 – 36 + 30}
k = 1
x2_{ + 5x + 1 = 7; x}2_{ + 5x + 1 - 7 = 0}
x2_{ + 5x – 6 = 0; factoring: ( x + 6 )( x – 1 ) = 0}
x = -6 and 1
answer: A
16. _{( 2x}2_{ + 11x – p ) / ( 2x – 3 )}
divisor: 2x – 3 = 0; 2x = 3; x = 3/2
using factor theorem,

2( 3/2 )2_{ + 11( 3/2 ) – p = 0}
2(9/4) + 33/2 – p = 0
9/2 + 33/2 – p = 0
42/2 – p = 0
p = 21
answer: D
17. _{y = -x + 3}
y = -x – 2

from the general form: y = mx + b, where the m is the slope. both equations have slopes equal to -1

* since the have the same slope, these lines are parallel.*
answer: B

18. _{2x – 3y = 12}

3x + y = 7

using substitution method:

3x + y = 7, y = 7 – 3x; then substitute to the other equation 2x – 3( 7 – 3x ) = 12 2x – 21 + 9x = 12 11x = 33 x = 3 y = 7 – 3x = 7 – 3(3) = 7 – 9 y = -2 solution: ( 3,-2 ) answer: A

19. _{let x – smaller number}

the sum of the two numbers is 125 x + x + 17 = 125

2x = 108

x = 54 --- smaller number answer: A

20. _{let x – number of ducks}

y – number of carabaos

there are 44 feet ( carabaos has 4 and ducks has 2 ) 2x + 4y = 44

there are 16 heads
x + y = 16, x = 16 – y
2( 16 – y ) + 4y = 44
32 – 2y + 4y = 44
2y = 44 – 32 = 12
y = 6 --- carabaos
answer: B
21. _{4x + 2 = 3x + 9}

where x is the number of students per row 4x – 3x = 9 – 2

x = 7, then substitute to above equation 4(7) + 2 = 30 --- students

answer: C

22. _{let x – adults}

then x – 289 is the number of children
there are a total of 737 persons, thus
x + x – 289 = 737
2x = 1026
x = 513 --- adults
children: x – 289 = 513 – 289 = 224
answer: C
23.
*h*
*d*

is the speed arriving 2 hrs late. Where d is the distance and h is time ( hrs.). To arrive in schedule, the train has to travel distance d in h – 2 hours.

Rate =

*time*

*ce*

*dis tan*

= *h*

###

### 2

*d*

answer: C
24. Ryan – 3 kph
Jerry – 2.4 kph
Ryan: ( 200m )( 1 hr / 3000m ) = 1/15 hr = 4min
Jerry: ( 200m )( 1 hr / 2400m ) = 1/12 hr = 5min
Difference: 1 min
answer: A
25. 180 miles in a 4-hour travel

1st_{ 3 hours: 50 mph}
( 50 m / hr )( 3 hr ) = 150 miles
180 – 150 = 30
30 miles in 1 hr
speed on the 4th_{ hr: 30 mph}
answer: A
26. let x – son
3x – man

Now 5 yrs from now

Son x x + 5

Man 3x 3x + 5

5 yrs from now, the man’s age is 3 more than twice the age of his son. 3x + 5 = 3 + 2( x + 5 ) 3x + 5 = 2x + 10 + 3 3x – 2x = 10 + 3 – 5 x = 8 man = 3x = 3(8) = 24 answer: B

27. let x = Jill’s age

x + 14 = Jack’s age

Now 10 yrs from now

Jill x x + 10

Jack x + 14 x + 24

x + 24 = 2( x + 10 ) x + 24 = 2x + 20

2x – x = 24 – 20 x = 4

Jack’s present age: x + 14 = 4 + 14 = 18 5 years from now: 18 + 5 = 23

answer: D 28. + = 30% 60% 50% 0.3( 10 ) + 0.6x = 0.5( 10 + x ) 3 + 0.6x = 5 + 0.5x 0.1x = 2 x = 20 L answer: C 29.

Rate Time Work

A 1/6 X x/6
B 1/4 X x/4
1
4
6
*x*
*x*

; multiply both sides by 12 2x + 3x = 12

5x = 12

x = 12/5 = 2 2/5 answer: B 30.

Rate Time Work

Grace 1/45 18 18/45
Abby 1/x 18 18/x
1
18
45
18 _{} _{}

*x* ; multiply both sides by 45x

18x + 18( 45 ) = 45x 45x – 18x = 810 27x = 810 x = 30 min answer: C 10 Liters x 10 + x