How Do You Solve the Modulus Inequality x|x - 2| ≥ 1?

[nyrychvantel]

For $$x|x - 2| \ge 1$$,

since $$|x - 2| \ge 0$$, and for $$x|x - 2| \ge 1$$,
so $$x > 0$$

Since $$x > 0$$, can I multiply the x inside to make it $$|{x^2} - 2x| \ge 1$$ ??


Or can you teach me your method of solving this question?
Thanks.

 

[praharmitra]

modulus function questions are usually solved by taking cases.

x > 2 and x < 2

Take the value of the mod according to the two cases and solve it.. After you solve the inequality for one case, you will get a bound for x. Don't forget to put this together with x > 2 to get the actual bound for x...

Accordingly do the next one

 

[Architeuthis Dux]

I can provide you with a logical and systematic approach to solving this modulus inequality question. Firstly, it is important to understand the concept of modulus, which is essentially the distance between a number and 0 on the number line. In this case, the modulus of |x - 2| represents the distance between x and 2 on the number line.

To solve this inequality, we need to consider two cases: when the expression inside the modulus is positive and when it is negative.

Case 1: When x - 2 is positive (x > 2)
In this case, the inequality becomes x(x - 2) \ge 1. Solving for x, we get x \ge 1 or x \ge 3.

Case 2: When x - 2 is negative (x < 2)
In this case, the inequality becomes -x(x - 2) \ge 1. Solving for x, we get x \le -1 or x \le 1.

Combining both cases, we get the solution set as x \le -1 or x \ge 1.

To answer your question, no, you cannot simply multiply the x inside the modulus to get |{x^2} - 2x| \ge 1. This is because the modulus function does not distribute over multiplication.

I hope this explanation helps you understand the concept and approach to solving modulus inequalities.