Question

||x|-1|<|x-1| solve for inequality ​

Answer 1

Topic :-

Inequalities

Given :-

||x| - 1| < |x - 1|

To Find :-

Values of 'x' satisfying the given inequality.

Solution :-

We solve this type of questions by making various cases.

So, we will be making various cases here also.

Inequality : ||x| - 1| < |x - 1|

Case I : x < 0, then

|-x - 1| < -x + 1

x + 1 < -x + 1

2x < 0

x < 0

Hence, x < 0.

Case II : x = 0, then

|0 - 1| < |0 - 1|

1 < 1

We will reject this case as 1 = 1.

Case III : 0 < x ≤ 1, then

|x - 1| < -x + 1

-x + 1 < -x + 1

We will reject this case as -x + 1 = -x + 1.

Case IV : x > 1

|x - 1| < x - 1

x - 1 < x - 1

We will reject this case as x - 1 = x - 1.

So, from the above results we can say that given inequality holds for x < 0 only.

Answer :-

The given inequality holds for x < 0.

Some Results to Remember :-

|x + y| = |x| + |y| when xy ≥ 0.

|x - y| = |x| + |y| when xy ≤ 0.

|x| > |y| implies x² > y².

||x| - |y|| < |x ± y| ≤ |x| + |y|