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.