Disprove the following statements using counterexample.
(a) |x| > 0
(b)√x2 = x
(c)√x2+4 = x+ 2
(d) x2 > x
(e) x ≤ x2
(a) The statement |x| > 0 is always true, so it cannot be disproven using a counterexample. For any real number x, the absolute value of x is always greater than zero.
(b) To disprove the statement √x^2 = x using a counterexample, we need to find a value of x for which the equation is not true. Let's choose x = -1. If we substitute this value into the equation, we get √(-1)^2 = -1, which simplifies to 1 = -1. Since this is not true, we have found a counterexample, disproving the statement.
(c) To disprove the statement √x^2 + 4 = x + 2 using a counterexample, we need to find a value of x for which the equation is not true. Let's choose x = 1. If we substitute this value into the equation, we get √1^2 + 4 = 1 + 2, which simplifies to √5 = 3. Since this is not true, we have found a counterexample, disproving the statement.
(d) To disprove the statement x^2 > x using a counterexample, we need to find a value of x for which the inequality is not true. Let's choose x = 0. If we substitute this value into the inequality, we get 0^2 > 0, which simplifies to 0 > 0. Since this is not true, we have found a counterexample, disproving the statement.
(e) The statement x ≤ x^2 is always true, so it cannot be disproven using a counterexample. For any real number x, x is always less than or equal to its square.
Sure, I can help you with that! To disprove the statements using counterexamples, we need to find specific values of x that contradict the given statement. Let's go through each statement one by one.
(a) |x| > 0:
This statement is actually always true. The absolute value of any number is always greater than zero. So, we cannot find a counterexample to disprove this statement.
(b) √x^2 = x:
For this statement to be false, we need to find a value of x for which the square root of x^2 is not equal to x. An example of a counterexample is x = -1. When we substitute x = -1 into the equation, we get √((-1)^2) = √1 = 1, which is not equal to -1.
(c) √(x^2 + 4) = x + 2:
To disprove this statement, we need to find a value of x that does not satisfy the equation. Let's try x = 0. When we substitute x = 0 into the equation, we get √(0^2 + 4) = √4 = 2, which is not equal to 0 + 2 = 2. Therefore, x = 0 is a counterexample that disproves the statement.
(d) x^2 > x:
For this statement to be false, we need to find a value of x for which x^2 is not greater than x. An example of a counterexample is x = 0. When we substitute x = 0 into the equation, we get 0^2 = 0, which is not greater than 0.
(e) x ≤ x^2:
To disprove this statement, we need to find a value of x that does not satisfy the inequality. Let's consider x = -1. When we substitute x = -1 into the equation, we get -1 ≤ (-1)^2, which simplifies to -1 ≤ 1. Since this is a true statement, we cannot find a counterexample to disprove the inequality.
In summary:
(a) |x| > 0 - This statement is always true and does not have any counterexamples.
(b) √x^2 = x - x = -1 is a counterexample that disproves this statement.
(c) √(x^2 + 4) = x + 2 - x = 0 is a counterexample that disproves this statement.
(d) x^2 > x - x = 0 is a counterexample that disproves this statement.
(e) x ≤ x^2 - This statement is always true and does not have any counterexamples.
|0| = 0
√x^2| = |x|. √(-2)^2 = √4 = 2
√(a^2 + b^2) ≠ a+b since (a+b)^2 = a^2+2ab+b^2
Why don't you try the others?