Test questions: Inequalities with no solution or all real numbers as solutions

📚 Quick Study Guide

• 🔢 When solving inequalities, you perform the same operations on both sides as you would with equations, with one crucial difference: if you multiply or divide both sides by a negative number, you must reverse the inequality sign.
• 🧮 An inequality has no solution when simplifying it leads to a contradiction, such as $5 > 5$ or $0 > 1$. In these cases, no value of the variable will make the inequality true.
• ✅ An inequality has all real numbers as solutions when simplifying it leads to an identity, such as $5 \le 5$ or $0 \le 1$. In these cases, any value of the variable will make the inequality true.
• 💡 To identify these special cases, solve the inequality as you normally would. If the variable cancels out and you are left with a true statement, all real numbers are solutions. If you are left with a false statement, there is no solution.
• 📝 Remember to distribute carefully and combine like terms to simplify the inequality before making a conclusion about its solution set.

Practice Quiz

1. Which of the following inequalities has no solution?
   1. $2x + 3 < 2x + 1$
   2. $3x - 5 > 3x - 7$
   3. $4x + 2 \le 4x + 2$
   4. $5x - 1 \ge 5x - 3$
2. Which of the following inequalities has all real numbers as solutions?
   1. $x + 5 > x + 3$
   2. $2x - 1 < 2x - 5$
   3. $3x + 4 \ge 3x + 4$
   4. $4x - 2 \le 4x - 6$
3. Solve the inequality: $3(x - 2) > 3x + 1$
   1. $x > 7$
   2. All real numbers
   3. No solution
   4. $x < -7$
4. Solve the inequality: $2(x + 1) \le 2x + 2$
   1. $x \le 0$
   2. All real numbers
   3. No solution
   4. $x \ge 0$
5. Which inequality has no solution?
   1. $4(x + 1) < 4x + 5$
   2. $5(x - 2) > 5x - 8$
   3. $6(x + 3) \le 6x + 18$
   4. $7(x - 1) \ge 7x - 7$
6. Which inequality has all real numbers as solutions?
   1. $8(x + 2) < 8x + 15$
   2. $9(x - 3) > 9x - 25$
   3. $10(x + 4) \le 10x + 40$
   4. $11(x - 5) \ge 11x - 54$
7. Solve: $-(2x - 1) \ge -2x + 3$
   1. No Solution
   2. All Real Numbers
   3. $x \ge 1$
   4. $x \le -1$