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