This is a tutorial on solving inequalities with absolute value. Detailed solutions and explanations are provided.
Example 1: Solve the inequality.
\( x + 2 \lt 3 \)
Solution to example 1

The above inequality is solved by writing a double inequality equivalent to the given inequality but without absolute value
\( 3 \lt x + 2 \lt 3 \)

Solve the double inequality to obtain
\( 5 \lt x \lt 1 \)

The above solution set is written in interval form as follows
\((5 , 1)\)
Example 2: Solve the inequality.
\( 2 x  4 \gt 9\)
Solution to example 2

Solving the above inequality is equivalent to solving
\(2 x  4 \gt 9\) or \(2 x  4 \lt  9\)

Which gives
\( x \lt 13 / 2\) or \( x \gt 5 / 2 \)

The above solution set is written in interval form as follows
\( (\infty , 13 / 2) \cup (5 / 2 , + \infty) \)
Example 3: Solve the inequality.
\( x + 2 \lt x^2  4 \)
Solution to example 3

Condition 1  For \( x^2  4 \gt = 0 \), or \( x\) in the interval \( (\infty , 2] \cup [2 , +\infty)\), we can write
\( x^2  4 = x^2  4 \)

Substitute the expression with the absolute value in the given inequality and solve
\( x + 2 \lt x^2  4\)
\( x^2  x  6 \gt 0\)

The solution set to above inequality is given by the interval
\( (\infty , 2) \cup (3 , +\infty)\)

The intersection of intervals \((\infty , 2] \cup [2 , +\infty)\) and \((\infty , 2) \cup (3 , +\infty)\) gives the solution set
\((\infty , 2) \cup (3 , +\infty)\)
Condition 2  For \(x^2  4 \lt 0\), or \(x\) in the interval \((2 , 2)\), we can write
\( x^2  4 = (x^2  4) \)

Substitute the expression with the absolute value in the given inequality and solve
\(x + 2 \lt (x^2  4)\)
\(x^2 + x  2 \lt 0\)

The solution set to above inequatlity is given by the intersection of the intervals \((2 , 1)\) and \((2 , 2)\) which gives
(2 , 1)

Conclusion: The solution set to the given inequality is
\((\infty, 2) \cup (2 , 1) \cup (3 , +\infty)\)
Check the above answer to the inequality graphically.(see graph below).
More references and links on how to Solve Equations, Systems of Equations and Inequalities.

