Exercise
Solve the inequality:
|x^2-1|\leq 1
Final Answer
Solution
|x^2-1|\leq 1
By absolute value definition, the inequality is equivalent to this inequality:
-1\leq x^2-1\leq 1
And this inequality is equivalent to these two the intersection between the inequalities:
-1\leq x^2-1 \text{ and } x^2-1\leq 1
Therefore, instead of solving the original inequality, we solve both inequalities and intersect their solutions. Let’s start with the first inequality:
-1\leq x^2-1
Solve the inequality:
0\leq x^2
We got an inequality that exists for every x, which means its solution is all x.
Moving on to the second inequality:
x^2-1\leq 1
Solve the inequality:
x^2-2\leq 0
It is a square inequality. Its roots are
x_{1,2}=-\sqrt{2}, \sqrt{2}
Therefore, the solution of the inequality is
-\sqrt{2}\leq x\leq \sqrt{2}
We intersect both solutions (“and”). That is, all x and
-\sqrt{2}\leq x\leq \sqrt{2}
The result of the intersection is
-\sqrt{2}\leq x\leq \sqrt{2}
And that’s the final answer.
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