Which of the following inequalities is true for all real values of x

Question

A: x^5=>x^4

B:5x^2=>2x^3

C:(5x)^2=>4x^2

D:3(x-2)^2=>3x^2-2

Answer 1

Hi Caleb, the answer is:

 

A:

x⁵ => x⁴

 

Divide both sides by x⁴:

 

x⁵ / x⁴ => 0

 

x => 0

 

False, if x is -1 the inequality is not true.

 

B:

 

5x² => 2x³

 

Divide both sides by 2x³

 

5x² / 2x³ => 0

 

x² will cancel out in both the numerator and denominator:

 

5  / 2x => 0

 

False, f x is -1 the inequality is not true.

 

C:  :

 

(5x)² => 4x²

 

Divide both sides by 4x²

 

(5x)² / 4 x^{2 => 0}

 

Remove the parenthesis:

 

 

5² x² / 4 x² => 0

 

x² cancels in both the numerator and denominator:

 

5² / 4 => 0

 

True for all real x.

 

D:  3(x-2)^2=>3x^2-2

 

3(x - 2)² => 3x² - 2

 

Add 2 to both sides:

 

3(x - 2)² + 2 => 3x²

 

Divide both sides by 3x²

 

(3(x - 2)² + 2) / 3x² => 0

 

If x = 0, 14 / 0 is undefined.

 

False, if x is 0, the inequality is false.

 

Questions?

 

 

Answer 2

Caleb, when in doubt, plug in some values for x and try them out.  If you have a graphing calculator you can easily compare the two equation

 

A Try x = .5   x⁵  =  .03125  while  x⁴ =.0626  whcih is large, not true

B  Try x =2  , 5x² = 20 while 2x³  = 16, true here  But when x =  4   5x² =  80  and 2x³ = 128 not true

C.  Is true because 5 is greater than 4 and the rest is the same and positive

D  Try x = 2  Then 3(x-2)²  = 0   but 3x² - 2  =  12  so not true