# Solutions and Explanations to Intermediate Algebra Questions in Sample 4

Full explanations of solutions for intermediate algebra questions in sample 4 are presented.

1. (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.

2. (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.

3. (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.

4. (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.

5. (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.

6. (True or False)   |x2 + 1| = x2 + 1.
Solution
Recall that if x > 0, then
| x | = x
Since x2 + 1 is positive for all values of real x, then
| x2 + 1 | = x2 + 1
The statement |x2 + 1| = x2 + 1" is TRUE.

7. (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.

8. (True or False)   (x - 2)(x + 2) = x2 - 4x - 4.
Solution
Expand (x - 2)(x + 2)
(x - 2)(x + 2) = x2 + 2x - 2x - 4
Group like terms
(x - 2)(x + 2) = x2 - 4
The statement "(x - 2)(x + 2) = x2 - 4x - 4" is FALSE.

9. (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.

10. (True or False)   |x - 3| = |x| + |3|, for all x real and negative.
Solution
|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.

11. (True or False)   (x + 2)3 = x3 + 23, for all x real.
Solution
Expand (x + 2)3
(x + 2)3 = (x + 2)(x + 2)2 = (x + 2)(x2 + 4 x + 4) = x3 + 6 x2 + 12 x + 8

12. (True or False)   If k = 4, then the equation x2 - k x = - 4 has one solution only.
Solution
Set k = 4 in the given equation
x2 - 4 x = -4
Solve it
x2 - 4 x + 4 = 0
(x - 2)2 = 0
one solution: x = 2

13. (True or False)   The discriminant of the equation: 2x2 - 4x + 9 = 0 is negative.
Solution
Calculate discriminant ?
? = (-4)2 - 4(2)(9) = 16 - 72 = -56
The discriminant ? is negative

14. (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) = (-x2 + 5 x - 6)(x - 4) = - x3 + 9 x2 - 26 x + 24
The degree of the given polynomial is the highest power and is therefore 3

15. (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

16. (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.

17. (True or False)   The relation 2y + x2 = 2 represents y as a function of x.
Solution
Solve the given equation for y
y = - x2 / 2 + 1
For each value of x we obtain one value of y only and therefore y is a function of x.

18. (True or False)   The relation 2y + x2 = 2 represents x as a function of y.
Solution
Solve the given equation for x
x2 = 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.

19. (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.

20. (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.

1. FALSE (different power)
2. TRUE
3. FALSE (1/(9-9) = 1 / 0 undefined)
4. TRUE
5. FALSE (|a - b| = a - b if b - a < 0 )
6. TRUE
7. FALSE √(x - 5) 2 = |x - 5|
8. FALSE
9. FALSE (try x = 16)
10. TRUE
11. FALSE (try x = 2)
12. TRUE
13. TRUE
14. FALSE (degree = 3)
15. FALSE (√2)
16. TRUE
17. TRUE
18. FALSE (solve for x and you get 2 solutions)
19. TRUE
20. TRUE

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