Absolute Values and Radicals Review

Miscellaneous

100

Simplify √(x²) for x∈R

|x|

200

Why does |f(x)|<0 have no solution?

Absolute values are non-negative by definition. Graphically there is no intersection.

300

What are the roots |x²-2x+1|?

x = 1 is the only root

400

When ordered from least to greatest value what is the correct order of: 

√(2),√(4/8),√(1),√(13)

√(4/8),√(1),√(2),√(13)

500

What are the restrictions for x in the following expression?

√(10-x)

x ≤10

Piecewise Notation

100

Write y=|3x-6| in piecewise notation

y = {3x-6 if x ≥ 3
      {-3x+6 if x < 3

200

Write |17x-18| in piecewise notation

y = {17x-18 if x≥18
      {-17x+18 if x<18

300

Write |-x+2| in piecewise notation

y = {x-2 if x≥2
      {-x+2 if x<2

400

write y=|x² - 6x + 8| in piecewise notation

y = {x² - 6x + 8 if x ≥ 4 or x ≤ 2
      {-x² + 6x - 8 if 2<x<4

500

Write y = |-x² + 5x - 4| in piecewise notation

y = {-x² + 5x - 4 if 1≤x≤4
      {x² -5x + 4 if x>4 or x<1

Absolute Value Equations

100

Solve |x-2|=4. State any real solution(s) and any extraneous solution(s).

x = -2, x = 6 are both real solutions.

200

Solve |x² - 6x + 2| = -1. State any real solution(s) and extraneous solution(s).

No real solutions exist

300

Solve |-2x-8| = 8x-9 graphically. 

x = 2.83 is a real solution

400

Solve |-x²-6x+9| = 20. State any real solution(s) and any extraneous solution(s). 

x = 3.16, x = -9.16

500

Solve |x²-2x+2|=-3x+4 graphically.

x = 1, x = -2 are real solutions

Rationalizing Denominators

100

Rationalize the denominator

__2__
7√(7)

  2√(7)  
    49

200

Rationalize the denominator and simplify

  √3 - √4 
    √12

 3 - 2√3 
    6

300

Rationalize the denominator and simplify

 √x-√2 
   √6x

  x√6 - 2√3x  
       6x

400

Rationalize the denominator and simplify

   2√7   
 √x - 7

  2√(7x) + 14√7  
     x² - 49

500

Rationalize the denominator and simplify

   √3-√(x-1)   
   √x + 8

   √(3x) - 8√(3) - √(x²-x) + 2√(2x-2)   
                 x - 64

Radical Equations

100

Solve √(3x) = 9 

x = 27

200

solve √(x-8) = 5

x = 33

300

Solve √(x) = √(3x-2)

State the real solution(s) and the extraneous solution(s) as exact values

x = 1 is a real solution

x = 1/2 is an extraneous solution

400

Solve x + √x - 6 = 7

State the real solutions and extraneous solutions

x = (27 - √53)/2 is a real solution
x = (27 + √53)/2 is an extraneous solution

500

To measure voltage in a 100 watt lightbulb, electrical engineers use the following formula:
V = √(100*R)
Where V is the voltage measured in volts, and R is the resistance measured in ohms.
What is the resistance of a 100 watt lightbulb plugged into a 120V outlet?

R = 144 ohms