Difference between revisions of "Trivial Inequality"
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For all [[real number]]s <math>x</math>, <math>x^2 \ge
For all [[real number]]s <math>x</math>, <math>x^2 \ge 0</math>.
Revision as of 19:12, 4 November 2021
The trivial inequality is an inequality that states that the square of any real number is nonnegative. Its name comes from its simplicity and straightforwardness.
For all real numbers , .
We can have either , , or . If , then . If , then by the closure of the set of positive numbers under multiplication. Finally, if , then again by the closure of the set of positive numbers under multiplication.
Therefore, for all real , as claimed.
The trivial inequality is one of the most commonly used theorems in mathematics. It is very well-known and does not require proof.
Suppose that and are nonnegative reals. By the trivial inequality, we have , or . Adding to both sides, we get . Since both sides of the inequality are nonnegative, it is equivalent to , and thus we have as desired.
- Find all integer solutions of the equation .
- Show that . Solution
- Show that for all real .
- Triangle has and . What is the largest area that this triangle can have? (AIME 1992)