Solve |x+1|>4, x in R.
Question
Solve $|x+1|>4, x \in \mathbb{R}$.
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Answer
:
Given: $$ |x+1| > 4 $$
The general property of absolute value inequalities is: $$ |x + b| > a \Leftrightarrow x + b < -a \ \text{or} \ x + b > a $$
Applying this property to our given inequality: $$ |x + 1| > 4 \Leftrightarrow x + 1 < -4 \ \text{or} \ x + 1 > 4 $$
Solving the two inequalities separately:
$$ x + 1 < -4 $$
Subtracting 1 from both sides: $$ x < -5 $$
$$ x + 1 > 4 $$
Subtracting 1 from both sides: $$ x > 3 $$
Thus, the solution to $|x+1| > 4$ is: $$ x < -5 \ \text{or} \ x > 3 $$
In interval notation, this can be expressed as: $$ (-\infty, -5) \cup (3, \infty) $$
Therefore, the solution set is: $$ (-\infty, -5) \cup (3, \infty) $$
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