Solve Inequalities With Absolute Value - Tutorials

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).

graph of x + 2 and |x^2 - 4| to check answer to inequality above.

More references and links on how to Solve Equations, Systems of Equations and Inequalities.