Solutions and Explanations
to Intermediate Algebra Questions in Sample 4
Full explanations of solutions for intermediate algebra questions in sample 4 are presented.

(True or False) x^{ 2} and 2 x are like terms.
Solution
The statement "x^{ 2} and 2 x are like terms" is FALSE because the two terms do not have the same power of x.

(True or False) x^{3} and 3x are unlike terms.
Solution
The statement "x^{3} and 3x are unlike terms" is TRUE because the two terms do not have the same power of x.

(True or False) 1 / (x  9) = 0 for x = 9.
Solution
Substitute x by 9 in the expression 1 / (x  9).
1 / (x  9) = 1 / (9  9) = 1 / 0 = undefined
The statement "1 / (x  9) = 0 for x = 9" is FALSE.

(True or False) The set of ordered pairs {(0,0),(2,0),(3,0),(10,0)} represents a function.
Solution
All values of the x coordinates are different and therefore the set of ordered pairs represents a function. The statement "The set of ordered pairs {(0,0),(2,0),(3,0),(10,0)} represents a function" is TRUE.

(True or False) a  b = b  a if b  a < 0.
Solution
Recall that if x > 0, then
 x  = x
But if b  a < 0, then a  b > 0 and
a  b = a  b
The statement "a  b = b  a if b  a < 0" is FALSE.

(True or False) x^{2} + 1 = x^{2} + 1.
Solution
Recall that if x > 0, then
 x  = x
Since x^{2} + 1 is positive for all values of real x, then
 x^{2} + 1  = x^{2} + 1
The statement x^{2} + 1 = x^{2} + 1" is TRUE.

(True or False) √(x  5) ^{2} = x  5.
Solution
For the above statement to be true, it must be true for all values of x for which the expression are defined. Let x =  4 and evaluate the right and left hand expressions.
Left: √(x  5) ^{2} = √(4  5) ^{2}
= √81 = 9
Right: x  5 =  4  5 =  9
The above statement is not true for x =  4 and therefore the statement "√(x  5) ^{2} = x  5" is FALSE.

(True or False) (x  2)(x + 2) = x^{2}  4x  4.
Solution
Expand (x  2)(x + 2)
(x  2)(x + 2) = x^{2} + 2x  2x  4
Group like terms
(x  2)(x + 2) = x^{2}  4
The statement "(x  2)(x + 2) = x^{2}  4x  4" is FALSE.

(True or False) √(x + 9) = √x + √9, for all x real.
Solution
For the above statement to be true, it must be true for all values of x for which the expression are defined. Let x = 16 and evaluate the right and left hand expressions.
Left: √(x + 9) = √(16 + 9)
= √(25) = 5
Right: √x + √9 = √(16) + √9
= 4 + 3 = 7
The above statement is not true for x = 16 and therefore the statement "√(x + 9) = √(x) + √(9), for all x real" is FALSE.

(True or False) x  3 = x + 3, for all x real and negative.
Solution
Start with x  3
x  3 = 3  x = 3 + ( x)
The absolute value of the sum of two positive numbers is equal to the sum of the numbers. Since x is negative  x is positive and hence
3 + ( x) = 3 + x = 3 + x
The statement "x  3 = x + 3, for all x real and negative" is TRUE.

(True or False) (x + 2)^{3} = x^{3} + 2^{3}, for all x real.
Solution
Expand (x + 2)^{3}
(x + 2)^{3} = (x + 2)(x + 2)^{2} = (x + 2)(x^{2} + 4 x + 4) = x^{3} + 6 x^{2} + 12 x + 8

(True or False) If k = 4, then the equation x^{2}  k x =  4 has one solution only.
Solution
Set k = 4 in the given equation
x^{2}  4 x = 4
Solve it
x^{2}  4 x + 4 = 0
(x  2)^{2} = 0
one solution: x = 2

(True or False) The discriminant of the equation: 2x^{2}  4x + 9 = 0 is negative.
Solution
Calculate discriminant ?
? = (4)^{2}  4(2)(9) = 16  72 = 56
The discriminant ? is negative

(True or False) The degree of the polynomial P(x) = (x  2)(x + 3)(x  4) is equal to 3.
Solution
Note that the degree of a polynomial is never negative. Expand the given polynomial
P(x) = (x  2)(x + 3)(x  4) = (x^{2} + 5 x  6)(x  4) =  x^{3} + 9 x^{2}  26 x + 24
The degree of the given polynomial is the highest power and is therefore 3

(True or False) The distance between the points (0 , 0) and (1 , 1) in a rectangular system of axes is equal to 1.
Solution
Distance between the given points is given by
√ ((1  0)^{2} + (1  0)^{2}) = √2

(True or False) The slope of the line 2x + 3y = 2 is negative.
Solution
Write the given equation in slope intercept form y = m x + b and identify the slope m.
3 y =  2x  2
y = (2/3) x  2/3
slope is equal to 2/3.

(True or False) The relation 2y + x^{2} = 2 represents y as a function of x.
Solution
Solve the given equation for y
y =  x^{2} / 2 + 1
For each value of x we obtain one value of y only and therefore y is a function of x.

(True or False) The relation 2y + x^{2} = 2 represents x as a function of y.
Solution
Solve the given equation for x
x^{2} = 2  2 y
x = ±√(2  2y)
For any value of y such that 2  2y ?0 we have two values of x and therefore x is not a function of y.

(True or False) The relation x = y DOES NOT represent x as a function of y.
Solution
Solve the given equation for x
x = ±y
For any value of y; we have two values of x and therefore x is not a function of y.

(True or False) The relation x = y DOES NOT represent y as a function of x.
Solve the given equation for y
y = ±x
For any value of x; we have two values of y and therefore y is not a function of x.
Answers to the Above Questions
 FALSE (different power)
 TRUE
 FALSE (1/(99) = 1 / 0 undefined)
 TRUE
 FALSE (a  b = a  b if b  a < 0 )
 TRUE
 FALSE √(x  5) ^{2} = x  5
 FALSE
 FALSE (try x = 16)
 TRUE
 FALSE (try x = 2)
 TRUE
 TRUE
 FALSE (degree = 3)
 FALSE (√2)
 TRUE
 TRUE
 FALSE (solve for x and you get 2 solutions)
 TRUE
 TRUE
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