Solved examples of absolute value inequalities |ax + b| > c problems

📚 Quick Study Guide

🔢 Definition of Absolute Value: The absolute value of a number $x$, denoted as $|x|$, is its distance from zero. Therefore, $|x| = x$ if $x \geq 0$, and $|x| = -x$ if $x < 0$.

💡 Solving Absolute Value Inequalities of the Form $|ax + b| > c$ :
• ➗ Split the inequality into two separate inequalities: $ax + b > c$ OR $ax + b < -c$.
• ✏️ Solve each inequality separately for $x$.
• 📈 The solution set is the union of the solution sets of the two inequalities.

🧭 Important Notes:
• 🛑 If $c$ is negative, and we have $|ax + b| > c$, the inequality is always true (assuming $ax+b \neq 0$).
• ✅ Remember to consider both positive and negative cases when dealing with absolute values.

Practice Quiz

1. What is the solution to the inequality $|x + 3| > 5$?
   1. $x > 2$ or $x < -8$
   2. $x > 2$ and $x < -8$
   3. $-8 < x < 2$
   4. $x < -2$ or $x > 8$
2. Solve the inequality $|2x - 1| > 7$.
   1. $x > 4$ or $x < -3$
   2. $x > 4$ and $x < -3$
   3. $-3 < x < 4$
   4. $x < -4$ or $x > 3$
3. What is the solution set for $|3x + 6| > 12$?
   1. $x > 2$ or $x < -6$
   2. $x > 6$ or $x < -2$
   3. $-6 < x < 2$
   4. $-2 < x < 6$
4. Find the solution to the inequality $|-x + 4| > 2$.
   1. $x < 2$ or $x > 6$
   2. $x > 2$ or $x < 6$
   3. $2 < x < 6$
   4. $x < -6$ or $x > -2$
5. Solve $|4x - 8| > 0$.
   1. $x \neq 2$
   2. $x > 2$
   3. $x < 2$
   4. All real numbers
6. Which of the following is the solution to $|5x + 10| > 15$?
   1. $x > 1$ or $x < -5$
   2. $x > 5$ or $x < -1$
   3. $-5 < x < 1$
   4. $-1 < x < 5$
7. Determine the solution for $|-2x - 4| > 6$.
   1. $x < -5$ or $x > 1$
   2. $x > -5$ or $x < 1$
   3. $-5 < x < 1$
   4. $x < -1$ or $x > 5$