Question 693458: Select the counterexample that makes this conjecture false:
For any real number x, x2 ≥ x.
x = |3|
x = -2
x = 0
x = 1/2
Found 2 solutions by stanbon, RedemptiveMath:
Answer by stanbon(75887)   (Show Source):
You can put this solution on YOUR website! Select the counterexample that makes this conjecture false:
For any real number x, x2 ≥ x.
x = |3|
x = -2
x = 0
x = 1/2
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Ans: x = 1/2 because (1/2)^2 = 1/4 < 1/2
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Cheers,
Stan H.
Answer by RedemptiveMath(80) (Show Source):
You can put this solution on YOUR website! A counterexample is an example that disproves a conjecture. It goes against what the conjecture states, and only one counterexample is needed to disprove a conjecture. The conjecture we are given is x^2 ≥ x. It wants to know out of the list of possible counterexamples, which one is actually a counterexample? Again, for one of these values of x to be a counterexample it must make the statement x^2 ≥ x false when we plug it in. So, we'll begin by plugging these values in for x:
|3|^2 ≥ |3|
3^2 ≥ 3
9 ≥ 3 (9 is greater than 3, so this is not a counterexample)
(-2)^2 ≥ (-2)
4 ≥ -2 (4 is greater than -2, so this is not a counterexample)
0^2 ≥ 0
0 ≥ 0 (0 is equal to 0, so this is not a counter example)
(1/2)^2 ≥ (1/2)
1/4 ≥ 1/2
The last example shows that when x = 1/2, the conjecture x^2 ≥ x is false because 1/4 is not greater than or equal to 1/2. This is what we would call a counterexample, and since we have found at least one counterexample, the conjecture x^2 ≥ x is not a sound conjecture. That is, it is not always true.
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