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.
To check your answers for the given inequalities, we can go through each option one by one and determine if they are true for all real values of x.
1) Inequality: x^2 ≤ x^3
To check if this inequality is true for all real values of x, we need to consider both positive and negative values of x. If we choose a positive value for x, such as x = 2, we get 4 ≤ 8, which is true. However, if we choose a negative value for x, such as x = -2, we get 4 ≤ -8, which is false.
Therefore, option a.) x^2 ≤ x^3 is not true for all real values of x.
2) Inequality: 2x^3 ≤ 4x^2
To check if this inequality is true for all real values of x, we can simplify it by dividing both sides by x^2 (assuming x ≠ 0). This gives us 2x ≤ 4. For any real value of x, the inequality 2x ≤ 4 is true. Therefore, option b.) 2x^3 ≤ 4x^2 is true for all real values of x.
3) Inequality: 4x^2 ≤ 9(4x)^2
To check if this inequality is true for all real values of x, we can simplify and solve it. Expanding the right side of the inequality gives us 4x^2 ≤ 144x^2. Simplifying further, we get -140x^2 ≤ 0. For positive values of x, this inequality is true. However, for negative values of x, it becomes false.
Therefore, option c.) 4x^2 ≤ 9(4x)^2 is not true for all real values of x.
4) Inequality: 4(x-3)^2 ≥ 4x^2 - 3
To check if this inequality is true for all real values of x, we can simplify and solve it. Expanding the left side of the inequality gives us 4x^2 - 24x + 36 ≥ 4x^2 - 3. Simplifying further, we get -24x + 36 ≥ -3. Rearranging terms, we have -24x ≥ -39, or dividing both sides by -24 (and reversing the inequality), we get x ≤ 39/24.
Therefore, option d.) 4(x-3)^2 ≥ 4x^2 - 3 is true for all real values of x.
So, your answer for the first question is incorrect. The correct answer is d.) 4(x-3)^2 ≥ 4x^2 - 3.
For the second question:
1) Inequality: (5x)^2 ≥ 5x^2
To check if this inequality is true for all real values of x, we can simplify and solve it. Expanding the left side of the inequality gives us 25x^2 ≥ 5x^2. This inequality is true for all positive values of x. However, for negative values of x, it becomes false.
Therefore, option a.) (5x)^2 ≥ 5x^2 is not true for all real values of x.
2) Inequality: x^6 ≤ x^7
To check if this inequality is true for all real values of x, we can simplify and solve it. Dividing both sides by x^6 (assuming x ≠ 0), we get 1 ≤ x. This inequality is true for all positive values of x, but for negative values of x, it becomes false.
Therefore, option b.) x^6 ≤ x^7 is not true for all real values of x.
3) Inequality: 5x^2 ≥ 2x^3
To check if this inequality is true for all real values of x, we need to consider both positive and negative values of x. If we choose a positive value for x, such as x = 2, we get 20 ≥ 16, which is true. However, if we choose a negative value for x, such as x = -2, we get 20 ≥ -16, which is also true.
Therefore, option c.) 5x^2 ≥ 2x^3 is true for all real values of x.
4) Inequality: 6(x-5)^2 ≥ 6x^2 - 5
To check if this inequality is true for all real values of x, we can simplify and solve it. Expanding the left side of the inequality gives us 6x^2 - 60x + 150 ≥ 6x^2 - 5. Simplifying further, we get -60x + 150 ≥ -5. Rearranging terms, we have -60x ≥ -155, or dividing both sides by -60 (and reversing the inequality), we get x ≤ 155/60.
Therefore, option d.) 6(x-5)^2 ≥ 6x^2 - 5 is true for all real values of x.
So, your answer for the second question is correct. The correct answer is d.) 6(x-5)^2 ≥ 6x^2 - 5.
I hope this helps!