Real Values of X (finding which inequality is true)

Question

I have a question... what is the answer to this problem? 

 

#1 Which of following inequalities is true for all real values of X? 

A.) x^2 ≤ x^3

B.) 2x^3 ≤ 4x^2

C.) 4x^2 ≤ (4x)^2

D.) 4(x-3)^2 ≥ 4x^2 -3

 

and 

 

#2 Which of the following inequalities is true for all real values of X? 

A.) (5x)^2 ≥ 5x^2

B.) x^6 ≤ x^7

C.) 5x^2 ≥ 2x^3

D.) 6(x-5)^2 ≥ 6x^2-5

 

 

 

Answer 1

Here's #1. Then I'll work on #2.

#1)

 

A) x^2<x^3, let x=-2, 4<-8 false

B) 2x^3<4x^2, x^3<2x^2, let x=3, 27<18 false

C) 4x^2<(4x)^2, 4x^2<16x^2, x^2<4x^2, 0<3x^2, 0<x^2 true for all real numbers because 0 is less than the square of any real number, even negative numbers because a negative times a negative is positive and also you have the symbol

" < " so if x=0 then 0<0 is a true statement

D) 4(x-3)^2>4x^2-3

    4(x^2-6x+9)>4x^2-3

    4x^2-24x+36>4x^2-3

    cancel both 4x^2 because there is one on each side

    -24x+36>-3

     -24x>-36-3

     -24x>-39

      24x<39

      let x=2, 48<39 false

 

2)

 

A) (5x)^2>5x^2, 25x^2>5x^2, 5x^2>x^2, 4x^2>0, x^2>0, true (similar to C above)

B) x^6<x^7, let x=-1, 1<-1 false

C) 5x^2>2x^3

    0>-5x^2+2x^3

    0>x^2(2x-5)

    2x-5<0

    2x<5

     x<2.5

    let x=3, 45>54, false

D) 6(x-5)^2>6x^2-5

    6(x^2-10x+25)>6x^2-5

     6x^2-60x+150>6x^2-5

      -60x+150>-5

      -60x>-155

       60x<155, let x=3, 180<155, false

   

Answer 2

Note that each choice has a x² and a x³ term.    We know that we when multiply an even number of negatives together, the result is positive.  When  we multiply an odd number of negatives together, the result is negative.  When we multiply just positives, the result remains positive.  Knowing this fact,

 

Use  x=-1 and x=1 to evaluate each choice.  Then compare the results.