# Solutions to Algebra Placement Test Practice

Solutions with explanations to algebra placement test practice.

1. Solution
Given expression
(a3 - 1) / (b - 1)
Substitute a by -2 and b by -2 in given expression
((-2)3 - 1) / ((-2) - 1)
Simplify expression above
(-8 - 1) / (-2 - 1) = 3

2. Solution
Given equation
- 2 (x + 9) = 20
Solve for x
- 2 x - 18 = 20
- 2 x = 38
Multiply both sides of the above equation
2( -2 x) = 2( 38)
Simplify to obtain
- 4 x = 76

3. Solution
x is a variable
20% of x is written as
20% x
the fifth of x is written as
(1 / 5) x
"20% of it is added to its fifth" is written as:
20% x + (1 / 5) x
seven tenths of x is written as
(7 / 10) x = 0.7 x
the result is equal to 12 subtracted from seven tenths of x
20% x + (1 / 5) x = 0.7x - 12
Change % and fractions to decimal numbers
0.2 x + 0.2 x = 0.7x - 12
Solve for x
12 = 0.3 x
x = 40

4. Solution
Given formula
A = 0.5 (b + B) h
Divide both sides of the formula by 0.5 h
(A / (0.5 h)) = 0.5 (b + B) h / (0.5 h)
simplify
(A / (0.5 h)) = b + B
Solve for B
B = (A / 0.5 h) - b
Note that 1 / 0.5 = 2, hence
B = 2 A / h - b

5. Solution
Let x be the amount of money in dollars that Tom has
Alex has the third of Tom
(1 / 3) x
Linda has twice as much as Alex
2 (1 / 3) x
All together they have 120 dollars, hence
x + x / 3 + 2 x / 3 = 120
Multiply all terms of the above equation by 3, simplify and solve for x
3 x + x + 2 x = 360
x = 60
Linda has
2 (1 / 3) x = (2 / 3) 60 = \$40

6. Solution
Given
6 x2 - 11 x - 2
The linear terms that will make the factors has coefficients 6 and 1 (since 6*1 = 6) or 2 and 3 (since 2*3=6)
(6x     )(x      )

(2x      )(3x     )
The constant terms should -1 and 2 or 1 and -2
(6 x - 1)(x + 2)
If (6x - 1)(x + 2) is expanded, it does not give 6 x2 - 11 x - 2.
(6 x - 1)(x + 2) = 6 x2 + 11 x - 2
We now try (6x + 1)(x - 2) which when expanded gives 6x2 - 11 x - 2
(6 x + 1)(x - 2) = 6 x2 - 11 x - 2

7. Solution
Given
x2 - 7 x - 8
Factor
(x + 1)(x - 8)
One of the listed factors is
x + 1

8. Solution
Given
(2xy2 - 3x2y) - (2x2y2 - 4x2y)
Eliminate brackets taking signs into consideration
= 2xy2 - 3x2y - 2x2y2 + 4x2y
Group like terms - 3x2y and 4x2y
= 2xy2  - 2x2y2 + x2y

9. Solution
Average speed is given by
Total distance / total time = 70 miles / hour
Total distance is given by
x + 200 miles
Total time is equal to
2 + 3 = 5 hours
Substitute in formula above
(x + 200) / 5 = 70
Solve above equation for x
x + 200 = 350
x = 150 miles

10. Solution
Rewrite all 4 equations in slope intercept form
(I) y = (- 3 / 2) x + 3 / 2
(II) y = (-2 / 3) x - 5 / 3
(III) y = (2 / 3) x - 3 / 2
(IV) y = - 3x + 9 / 2
The slopes of all 4 lines are
(I) -3/2
(II) -2/3
(III) 2/3
(IV) -3
For two lines with slopes m and n to be perpendicular, the product of their slopes must equal -1 or m*n = -1.
The product of the slopes of equations (I) and (III) is given by

(-3 / 2)*(2 / 3) = - 1
hence (I) and (III) are perpendicular.

11. Solution
Given
f (x) = (x + 1)2
To find f (t + 2), we substitute x by t + 2 in f (x); hence
f (t + 2) = ((t + 2)+ 1) 2
then simplify and expand
= (t + 3) 2
= t 2 + 6t + 9

12. (√x + √y) (√x - √y) - (√x - √y)2 =

Solution
Given
(√x + √y) (√x - √y) - (√x - √y) 2
Expand (√x + √y) (√x - √y)
(√x + √y) (√x - √y) = (√x) 2 - (√y) 2 = x - y
Expand (√x - √y)2
(√x - √y)2 = (√x) 2 + (√y) 2 - 2√x √y = x + y - 2√x √y
Hence
(√x + √y) (√x - √y) - (√x - √y) 2
= (x - y) - (x + y - 2√x √y)
= - 2 y + 2√x √y

13. x / 2 - y / 4 = 7

Solution
Equation of line given in standard form
x / 2 - y / 4 = 7
To determine the slope, we need to write the given equation in slope intercept form. Multiply all terms by 4 and simplify and solve for y.
2 x - y = 28
y = 2x - 28
Slope is the coefficient of x and is equal to
2

14. (x / (x - 3) + 1 / 2)(2/(x - 1)) =

Solution
Given
(x / (x - 3) + 1 / 2) (2 / (x - 1))
We first simplify the expression ( x / (x - 3) + 1 / 2). Reduce to the lowest common denominator.
( x / (x - 3) + 1 / 2)
2x / ( 2 (x - 3) ) + ( x- 3)  / (2 (x - 3))
= (3x - 3) / (2(x - 3))
Hence
x / (x - 3) + 1 / 2) (2 / (x - 1))

= (3x - 3) (2/(x - 1))) / (2(x - 3))
= 3(x - 1)(2 / (x - 1)) / (2(x- 3))
Simplify
= 3 / (x - 3)

15. Solution
Let a and b be the x and y coordinates of point B, hence
M is the midpoint of A and B
Since the midpoint M(3,2) is known and point A(2,1) is also known, we can write
(3,2) = ( (2 + a)/2 , (1 + b)/2 )
Two equations may be written as follows
3 = (2 + a) / 2 and 2 = (1 + b) / 2
Solve for a and b
a = 4 and b = 3 |>