It doesn't. If we squared both sides and did it mechanically we'd get \(\displaystyle 1 \leq cos^2(2x) \leq 1\), which is a ridiculous statement. (It is unwise to square both sides of an inequality. Weird things happen.) Instead we set the lower bound by noting that the square of any real quantity is greater than or equal to 0.tmt said:I have
$$-1 \le \cos\left({2x}\right) \le 1 $$
If everything is squared, it goes to
$$0 \le \cos^2\left({2x}\right) \le 1 $$
and I'm not sure how $(-1)^2$ turns into $0$
tmt said:I have: $$-1 \le \cos\left({2x}\right) \le 1 $$
If everything is squared, it goes to: $$0 \le \cos^2\left({2x}\right) \le 1 $$
and I'm not sure how $(-1)^2$ turns into $0$. . . Actually, it doesn't.
Simplifying an inequality means to manipulate or rearrange an inequality expression in order to make it easier to understand or solve.
Simplifying an inequality allows us to clearly see the relationship between the variables in the expression and make it easier to determine the solution or graph the inequality.
Some common strategies for simplifying an inequality include combining like terms, distributing, and isolating the variable on one side of the inequality symbol.
To check your solution, you can substitute the value you found for the variable back into the original inequality and see if it still holds true.
Some common mistakes to avoid when simplifying an inequality include forgetting to flip the inequality symbol when multiplying or dividing by a negative number, making arithmetic errors, and not checking the solution.