Can someone please check my answers? Thanks! :)

1.) Which of the following inequalities is true for all real values of x?

a.) x^2≤x^3
b.) 2x^3≤4x^2
c.) 4x^2≤94x)^2
d.) 4(x-3)^2≥4x^2-3

My answer: B.

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

My answer: D.

1

a
try x = -1
1 </= -1 nope

b
x = -1 works
but x = 10 fails totally

c
x^2 </= 23.25 x^2
x^2 always +
so this works

by the way d is

4 x^2 -24 x + 36 >/= 4 x^2 - 3

-24 x + 39 >/= 0
nope

To check your answers for these inequalities, we can simplify each option and see which inequality holds true for all real values of x.

1.) For option a:

x^2 ≤ x^3

To simplify, let's divide both sides by x^2 (assuming x ≠ 0):

1 ≤ x

This inequality is not true for all real values of x, so option a is incorrect.

2.) For option b:

2x^3 ≤ 4x^2

To simplify, let's divide both sides by 2x^2 (assuming x ≠ 0):

x ≤ 2

This inequality is true for all real values of x, so option b is correct.

3.) For option c:

4x^2 ≤ 94x)^2

There seems to be a typographical error in the expression. It should be 94x^2 rather than 94x)^2. Since we cannot interpret the expression as it is, we cannot evaluate its truthfulness. Therefore, option c is invalid.

4.) For option d:

4(x-3)^2 ≥ 4x^2-3

Expanding the left side:

4(x^2 - 6x + 9) ≥ 4x^2 - 3

Simplifying:

4x^2 - 24x + 36 ≥ 4x^2 - 3

Rearranging:

-24x + 36 ≥ -3

-24x ≥ -39

Dividing by -24 (Note: when you multiply or divide an inequality by a negative number, you must reverse the inequality symbol):

x ≤ 39/24

This inequality is not true for all real values of x, so option d is incorrect.

Based on this analysis, your answer for question 1 is correct (B).

Now let's move on to question 2:

1.) For option a:

(5x)^2 ≥ 5x^2

Expanding:

25x^2 ≥ 5x^2

Dividing by 5x^2 (assuming x ≠ 0):

5 ≥ 1

This inequality is true for all real values of x, so option a is correct.

2.) For option b:

x^6 ≤ x^7

Dividing by x^6 (assuming x ≠ 0):

1 ≤ x

This inequality is true for all real values of x, so option b is correct.

3.) For option c:

5x^2 ≥ 2x^3

We cannot simplify this inequality further. However, let's consider a counterexample for x = 1:

5(1)^2 ≥ 2(1)^3

5 ≥ 2

This inequality is true, so it holds for x = 1. However, if we choose x = -1, the inequality does not hold:

5(-1)^2 ≥ 2(-1)^3

5 ≥ -2

This inequality is not true. Therefore, option c is incorrect.

4.) For option d:

6(x-5)^2 ≥ 6x^2 - 5

Expanding the left side:

6(x^2 - 10x + 25) ≥ 6x^2 - 5

Simplifying:

6x^2 - 60x + 150 ≥ 6x^2 - 5

Rearranging:

-60x + 150 ≥ -5

-60x ≥ -155

Dividing by -60 (Note: when you multiply or divide an inequality by a negative number, you must reverse the inequality symbol):

x ≤ 155/60

This inequality is true for all real values of x, so option d is correct.

Based on this analysis, your answer for question 2 is also correct (D).

Great job! Your answers for both questions are correct.