Solutions and Explanations to Intermediate
Algebra Questions in Sample 3
Solutions with full explanations of the intermediate algebra questions in sample 3 are presented.

Write 1.5 × 10^{5} in standard form.
Solution
1.5 × 10^{5} = 1.5 / 10^{ 5} = 1.5 / 100000 = 0.000015

Evaluate: 30  x + 6 for x = 10
Solution
Substitute x by 10 in the given expression and evaluate
30   (10) + 6 = 30  10 + 6
= 30    4  = 30  4 = 26

Evaluate: 2xy^{3} + x  2y for x = 2 and y = 2
Solution
Substitute x by 2 and y by  2 in the given expression and evaluate
2(2)(2)^{3} + 2  2(2)
= 4(8) + 2 + 4 =  26

What is the slope of the line perpendicular to the line y =  4
Solution
y =  4 is a horizontal line and any line perpendicular to it is vertical and therefore has an undefined slope.

Write an equation of the line with slope 2 and xintercept ( 4 , 0).
Solution
A line with slope m and passes through point (a , b) has an equation of the form.
y  b = m(x  a) , point slope form of a line
Use the given slope and the point to write the equation of the line.
y  0 = 2(x  (4))
y = 2x + 8

Solve the equation: 3(x + 5) + 20 = 10(x  3) + 4
Solution
Expand the expressions with brackets in the given equation
3x  15 + 20 = 10x + 30 + 4
Group like terms
3x + 5 = 10x + 34
Solve
13 x = 29 , x = 29 / 13

Solve the inequality: 4(x  6) + 4 < 8(x  4)
Solution
Expand the expressions with brackets in the given inequality
4x  24 + 4 < 8x  32
Group like terms
4x  20 < 8x  32
Solve the inequality
4x  8x < 32 + 20
4x < 12
x > 3

Solve the equation: 3(x  2)^{2}  12 = 0
Solution
Rewrite equation with square on one side
3(x  2)^{2} = 12
(x  2)^{2} = 4
x  2 = (+ or ) 2
x = 2 + 2 = 4 , x = 2  2 = 0
The solutions of the given equation are
x = 4 and x = 0

Solve the equation: x / 3 + 2 / 7 = x / 7  5
Solution
The given equation has denominators that need to be cleared by multiplying all terms of the equation by the lowest common denominator of the denominators 3 and 7 which is 21
21(x / 3 + 2 / 7) = 21(x / 7  5)
Expand and simplify
7x + 6 = 3x  105
Solve for x
4x =  111 , x =  111/4

Line L is defined as line through the point (2 , 7) and perpendicular to the line x + y = 0. What is the point of intersection of L and the line x + y = 0?
Solution
We first find the slope of the line x + y = 0. Rewrite in slope intercept form
y =  x , slope =  1
Line L is perpendicular to line x + y = 0 and therefore the product of its slope m and the slope of the line x + y = 0 is equal to 1. Hence
m*(1) = 1 , solve for m: m = 1
Let A(a , b) be the point of intersection. Point A and the given point (2 , 7) lie on line L and therefore the slope calculated using point (a , b) and (2 , 7) must be equal to the slope of line L which is  1. Hence
(7  b) / (2  a) = 1
Point A(a , b) lie also on line x + y = 0. Hence
a + b = 0
Solve the two system of equations (7  b) / (2  a) = 1
and a + b = 0 to find point A(a , b).
a + b = 0 implies that a =  b
Cross multiply (7  b) / (2  a) = 1 to obtain
7  b = 2  a
Substitute a by  b (a =  b above) in the above equation and solve
7  b = 2  (b) , b = 5 / 2
Use a =  b (see above) to find a
a =  5/2
The point of intersection is
A( 5/2 , 5/2)

What is the point of intersection of the lines: x + 2y = 4 and x  3y = 7?
Solution
Since the point of intersection lie on the two lines, its coordinates x and y satisfy the two equations simultaneously and are therefore found by solving the system of equations of the two lines. Let us add the right hand side and left hand side of the two equations
x + 2y = 4
+
 x  3y = 7
 y =  3 , equation obtained
y = 3
Substitute y by 3 in one of the equations to find x.
x + 2(3) = 4 , solve for x: x =  2
The point of intersection is given by its coordinate as follows
(2 , 3)

How many solutions do the system of equations 2x  3y = 4 and 4x  6y = 7 have?
Solution
Multiply all terms of the equation 2x  3y = 4 to obtain
4 x  6 y = 8
Subtract right sides and left sides the equation obtained above from the equation 4x  6y = 7
(4 x  6 y)  (4x  6y) = 8  (7)
Simplify
0 = 15
The above statement is false and therefore the system of equations has no solution.

For what value(s) of A does the system of equation A x + 6y = 0 and 2x  7y = 3 have no solutions?
Solution
Solve equation A x + 6y = 0 for y
y =  A x / 6
Substitute y by  A x / 6x into the equation 2x  7y = 3
2 x  7( A x / 6) = 3
Solve the above for x
x = 3 / (2 + 7 A / 6)
x is a not a solution if the denominator 2 + 7 A / 6 is equal to zero
2 + 7 A / 6 = 0
Solve for A
A =  12 / 7

Solve 2x  4  2 = 6.

How many solutions does the equation 2x^{2} + 3x = 8 have?

Solve the equation 3x^{2} + 6x  1 = 8.

Solve the system of equations: 2x + 5y = 18 and 3x  y = 1.

What is the range of function f defined by: f = {(2,3),(1,4),(5,4),(0,3)}

Factor the expression 2x^{2} + 3x + 1.

Factor the expression 10x^{2} + 20x  80.
Answers to the Above Questions
 0.000015
 26
 26
 undefined (vertical line)
 y = 2x + 8
 29/13
 x > 3
 {0 , 4}
 111/4
 (5/2 , 5/2)
 (2 , 3)
 no solutions
 12/7
 {2 , 6}
 2 real solutions
 {3 , 1}
 (1 , 4)
 {3 , 4}
 (2x + 1)(x + 1)
 10(x  2)(x + 4)
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