Which of the following equivalences is true?
x^2 ≥ 0 ⇔ (x+1)^2 ≥ 1
x^2 ≥ 0 ⇔ x^2+1 ≥ 1
x > 0 ⇔ x^2 > 0
x^2 > 0 ⇔ x^2+1 ≥ 1
x ≥ 0 ⇔ x^2 ≥ 0
Looks like all of them are true.
Yes, this is what I thought!! Im so confused unless this is a trick question.
To determine which of the given equivalences is true, let's analyze each option step by step:
1. x^2 ≥ 0 ⇔ (x+1)^2 ≥ 1:
To check if this equivalence is true, we need to simplify both sides of the equation.
On the left-hand side, x^2 is non-negative for all real values of x, so x^2 ≥ 0 is always true.
On the right-hand side, we have (x+1)^2 which is also non-negative for all real values of x. Since (x+1)^2 ≥ 1, this equivalence is true.
2. x^2 ≥ 0 ⇔ x^2+1 ≥ 1:
To verify this equivalence, we need to simplify both sides.
The left-hand side is x^2 ≥ 0, which is always true since x^2 is non-negative for all real values of x.
However, on the right-hand side, x^2+1 ≥ 1 is not true for all values of x. If x = 0, the inequality becomes 1 ≥ 1, which is true. But if x = -1, the inequality becomes 0 ≥ 1, which is false.
Therefore, this equivalence is not true.
3. x > 0 ⇔ x^2 > 0:
To determine if this equivalence holds, let's analyze both sides.
The left-hand side states that x is strictly greater than 0. This means x must be a positive number.
On the right-hand side, we have x^2 > 0. Since x^2 is always non-negative, x^2 > 0 is true for all positive values of x.
Therefore, this equivalence is true.
4. x^2 > 0 ⇔ x^2+1 ≥ 1:
To evaluate this equivalence, let's simplify both sides.
The left-hand side is x^2 > 0, which is true for all non-zero values of x since x^2 is positive when x is not zero.
On the right-hand side, x^2+1 ≥ 1 always holds since x^2 + 1 is always greater than or equal to 1.
Therefore, this equivalence is true.
5. x ≥ 0 ⇔ x^2 ≥ 0:
For this equivalence, we need to analyze both sides.
The left-hand side states that x is greater than or equal to 0. In other words, x can be either positive or zero.
On the right-hand side, x^2 ≥ 0 is always true because the square of any real number is non-negative.
Therefore, this equivalence is true.
Summarizing the results:
- Options 1, 3, 4, and 5 are all true.
- Option 2 is not true.
So, the correct answer is: x^2 ≥ 0 ⇔ (x+1)^2 ≥ 1.