1. Radicals and Inequalities
Practice Quiz
Radicals and Inequalities

2. 7 + 4 11 Find x + 4 when x = 7 4x – 3 = 25 +3 +3 4x = 28 x = 7
If , then find
the value of x + 4 ? 1
Find x + 4
when x = 7
7 + 4
4x – 3 = 25 +3 +3 11
4x = 28
x = 7

3. If , then x – 5 equals 2
Find x – 5
when x = 18 –7 –7
18 – 5 13
2x = 36
x = 18

4. 49 < x – 2 < 64 51 < x < 66 +2 +2 +2
If x is an integer and , how
many different values of x are possible? 3
First, solve
inequality.
49 < x – 2 < 64 +2 +2 +2
51 < x < 66

5. 51 < x < 66 If x is an integer and , how 3
many different values of x are possible? 3
51 < x < 66
Integers between
51 and 66 OR
66 – 51 = 15
15 – 1 = 14
Answer: 14 Values

6. 3 Try this strategy How many values of m are possible?
Use an inequality with two numbers
that are close together.
2 < m < 7
Integers between 2 and 7
How many values
of m are possible?
10 < m < 45
Answer: 4 Values OR
45 – 10 = 35
35 – 1 = 34
7 – 2 = 5
5 – 1 = 4

7. 9 < m + 1 < 25 8 < m < 24 –1 –1 –1
If m is an integer and , how
many different values of m are possible? 4
First, solve
inequality.
9 < m < 25 –1 –1 –1
8 < m < 24

8. 8 < m < 24 If m is an integer and , how 4
many different values of m are possible? 4
8 < m < 24
Integers between
8 and 24 OR
24 – 8 = 16
16 – 1 = 15
Answer: 15 Values

9. 5
5x + 3 = 3x + 7
–3x
–3x
2x + 3 = –3 –3
2x =
x =

10. 6
The equation has ___ solution(s).
(x + 5)2
(x + 5)(x + 5)
x2 + 5x + 5x + 25
2x + 13 = x2 + 10x + 25
x2 + 10x + 25
–2x – 13
–2x – 13
0 = x2 + 8x + 12 12
0 = (x + 2)(x + 6) 2 6
x + 2 = 0 ,
x + 6 = 0 8
x = –2 ,
x = –6

11. 3 = 3 1  –1 6 x = –2 x = –6 The equation has ___ solution(s). YES NO
Check Answers
x = –2
x = –6
3 = 3
1  –1
One negative solution
YES NO

12. x – 2 = x2 – 4x + 4 –x + 2 –x + 2 0 = x2 – 5x + 6 0 = (x – 2)(x – 3)7
The equation has ___ solution(s).
(2 – x)2
(2 – x)(2 – x)
4 – 2x – 2x + x2
x – 2 = x2 – 4x + 4
x2 – 4x + 4
–x + 2
–x + 2
0 = x2 – 5x + 6 6
0 = (x – 2)(x – 3) –2 –3
x – 2 = 0 ,
x – 3 = 0 –5
x = 2 ,
x = 3

13. 0 = 0 1  –1 7 x = 2 x = 3 The equation has ___ solution(s). YES NO
Check Answers
x = 2
x = 3
0 = 0
1  –1
YES NO
One positive solution

14. 8
–25
–25
w = 256

15. If and , then what is the
value of b ? 9
a = 64
b = 16

16. y Step 1 Step 2 Step 3 y = 16 x = 9 10 If and , then what is the
value of x ? 10
Step 1
Step 2
Step 3 y
y = 16
x = 9

17. 11
0 > 5x – 1 +1 +1
1 > 5x

18. 12
–3x + 13 < –14
–13
–13
–3x < –27
x > 9

19. 13
7x – 3 > 4x – 5
–4x
–4x
3x – 3 > –5 +3 +3
3x > –2

20. 14 2x = 1 NO NO If , then which of the
following statements must be true? 14
Strategy: Substitute ¼ for x in each answer.
2x = 1 NO NO

21. 14 2x > 1 NO NO If , then which of the
following statements must be true? 14
Strategy: Substitute ¼ for x in each answer.
2x > 1 NO NO

22. 15 2x = –1 YES NO NO If –1 < x < 0 , then which of the
following statements must be true? 15
Strategy: Substitute –½ for x in each answer.
2x = –1
YES NO NO

23. 15 2x > 1 NO YES If –1 < x < 0 , then which of the
following statements must be true? 15
Strategy: Substitute –½ for x in each answer.
2x > 1 NO
YES

24. If –1 < x < 0 , then which of the
following statements must be true? 15
Strategy: Substitute –⅛ for x in each answer.
A. 2x = –1
is not correct NO

25. If , then which of the following inequalities is correct?16
m = 0.5
m3 = 0.125
m3 < m2 < m

26. If , then which of the following inequalities is correct?17
K = –0.333
K3 = –0.037
K < K3 < K2

27. 10 < x x 10 < x x –10 10 < –10 –5 10 < –5 –1 < –10
If , then which of the following values could be x ? 18
Strategy: Test each answer by substituting for x.
A. –10
B. –5 10
< x x 10
< x x
–10 10
< –10 –5 10
< –5
–1 < –10
–2 < –5 NO NO

28. 10 < x x 10 < x x 10 < x x 1 10 < 1 2 10 < 2 5 10
If , then which of the following values could be x ? 18
C. 1
D. 2
E. 5 10
< x x 10
< x x 10
< x x 1 10
< 1 2 10
< 2 5 10
< 5
10 < 1
5 < 2
2 < 5 NO NO
YES

29. ½ > 3 x > x 6 x > x 6 6 > 6 6 3 > 3 1 > 6 A. 6 B. 3
If , then which of the following values could be x ? 19
Strategy: Test each answer by substituting for x.
A. 6
B. 3 x
> x 6 x
> x 6 6
> 6 6 3
> 3
1 > 6 NO
½ > 3 NO

30. –½ > –3 x > x 6 x > x 6 6 > 0 6 –3 > –3 0 > 0 C. 0
If , then which of the following values could be x ? 19
C. 0
D. –3 E. x
> x 6 x
> x 6
Cannot be
determined 6
> 0 6 –3
> –3
0 > 0
–½ > –3 NO
YES

31. z < 0 20  y = ( + ) positive y > 0 x > y 
If x > y and y > 0 and xz < 0, then which
of the following must be true about all the values of z? 20 
y = ( + ) positive
y > 0
x > y 
x = ( + ) positive
xz < 0 
xz = ( – ) negative
(+)z = ( – )
z < 0
(+)(–) = ( – )
z is negative

32. x = y m > y m > x x < m 21
If x = y and m > y, then which of the following must be true? 21
I. x < m II. x = m III. x > m
x = y
(Given)
m > y
m > x
(Given)
Equivalent to
x < m

33. 22 xy < 0  xy = negative answer   x > y x is positive +
If x > y and xy < 0, which of the following inequalities must be true? 22
I. x > 0 II. y > 0 III.
xy < 0 
xy = negative answer
One number is positive
One number is negative  
x > y
x is positive +
y is negative –

34. 22  x is positive + x > y y is negative –
If x > y and xy < 0, which of the following inequalities must be true? 22
I. x > 0 II. y > 0 III. 
x > y
x is positive +
y is negative –

35. x – 7 = 0 x – 1 = 0 x = 7 x = 1 1 7 (0 – 7)(0 – 1) > 0
What are all the values of x for which
(x – 7)(x – 1) > 0 ? 23
x – 7 = 0
x – 1 = 0
x = 7
x = 1 A B C
YES NO
YES 1 7
Test Section A
with x = 0
(0 – 7)(0 – 1) > 0
(–7)(–1) > 0
x < 1 or x > 7
7 > 0
YES

36. x + 6 = 0 x + 3 = 0 x = –6 x = –3 –6 –3 (0 + 6)(0 + 3) < 0
What are all the values of x for which
(x + 6)(x + 3) <0 ? 24
x + 6 = 0
x + 3 = 0
x = –6
x = –3 A B C NO
YES NO –6 –3
Test Section C
with x = 0
(0 + 6)(0 + 3) < 0
(6)(3) < 0
–6 < x < –3
18 < 0 NO

37. x + 2 = 0 x – 5 = 0 x = –2 x = 5 –2 5 (0 + 2)(0 – 5) > 0
What are all the values of x for which
(x + 2)(x – 5) > 0 ? 25
x + 2 = 0
x – 5 = 0
x = –2
x = 5 A B C
YES NO
YES –2 5
Test Section B
with x = 0
(0 + 2)(0 – 5) > 0
(2)(–5) > 0
x < –2 or x > 5
–10 > 0 NO