Solve inequality: sqrt(x^2) + x < 1

There are two possible solutions, substitution and using absolute value. Here, we will use absolute value method.By definition of square root we get: sqrt(x^2) = |x|. By definition of absolute value we get:|x| = x, if x >= 0 and |x| = -x, if x<0 (Definition 1). Rewrite inequality as: |x| + x < 1 Thus, we need to observe two possible options, where x is non-negative and negative.

1. x >= 0, thus x + x < 1 (By Definition 1) x >= 0, 2x < 1 x>= 0 , x < 1/2 The range for this system is [0; 1/2)

2. x < 0, thus -x + x < 1 0 < 1 (always true) The range for this system is (-infinity, 0)

We combine these two intervals and get final answer (-infinity, 1/2)