Mathematical problems on inequalities with answers and steps:Multiple
Choice Questions on Inequalities: Linear and Nonlinear
1. Solve the inequality: 2x + 5 > 13
A) x > 4
B) x > 5
C) x < 4
D) x < 5
Answer: A) x > 4
Steps:
2x + 5 > 13
2x > 8
x > 4
2. Which of the following is the solution set for |x + 2| ≤ 3?
A) [-5, 1]
B) [-1, 5]
C) [-5, -1]
D) [1, 5]
Answer: B) [-1, 5]
Steps:
|x + 2| ≤ 3
-3 ≤ x + 2 ≤ 3
-5 ≤ x ≤ 1
Therefore, x ∈ [-1, 5]
3. Solve the compound inequality: -2 < 3x - 1 ≤ 5
A) 1 < x ≤ 2
B) -1 < x ≤ 2
C) 1 ≤ x < 2
D) -1 ≤ x < 2
Answer: A) 1 < x ≤ 2
Steps:
-2 < 3x - 1 ≤ 5
-1 < 3x ≤ 6
-1/3 < x ≤ 2
1 < x ≤ 2
4. Which inequality represents the shaded region?
[Image of a number line with shading to the left of -3]
A) x < -3
B) x > -3
C) x ≤ -3
D) x ≥ -3
Answer: A) x < -3
5. Solve: x² - 4x - 5 < 0
A) -1 < x < 5
B) x < -1 or x > 5
C) -1 < x < 5
D) x < -1 and x > 5
Answer: A) -1 < x < 5
Steps:
Factoring: (x - 5)(x + 1) < 0
The inequality is satisfied when one factor is negative and the other is positive.
x - 5 < 0 and x + 1 > 0, or
x < 5 and x > -1
Therefore, -1 < x < 5
6. Which of the following is equivalent to x > 2 and x < 5?
A) 2 < x < 5
B) x > 2 or x < 5
C) x ≥ 2 and x ≤ 5
D) 2 ≤ x < 5
Answer: A) 2 < x < 5
7. Solve: 2(x - 3) ≥ 4x + 2
A) x ≤ -4
B) x ≥ -4
C) x ≤ 4
D) x ≥ 4
Answer: A) x ≤ -4
Steps:
2x - 6 ≥ 4x + 2
-2x ≥ 8
x ≤ -4
8. What is the solution set of |2x - 1| > 5?
A) x < -2 or x > 3
B) -2 < x < 3
C) x < 3 or x > -2
D) x < -2 and x > 3
Answer: A) x < -2 or x > 3
Steps:
|2x - 1| > 5
2x - 1 < -5 or 2x - 1 > 5
2x < -4 or 2x > 6
x < -2 or x > 3
9. Solve: 3x + 2 < x - 4
A) x > 3
B) x < -3
C) x > -3
D) x < 3
Answer: B) x < -3
Steps:
3x + 2 < x - 4
2x < -6
x < -3
10. Which inequality represents the solution to 5 - 2x ≤ 1?
A) x ≥ 2
B) x ≤ 2
C) x ≥ -2
D) x ≤ -2
Answer: A) x ≥ 2
Steps:
5 - 2x ≤ 1
-2x ≤ -4
x ≥ 2
11. Solve: x²/4 - 1 > 0
A) x < -2 or x > 2
B) -2 < x < 2
C) x ≤ -2 or x ≥ 2
D) -2 ≤ x ≤ 2
Answer: A) x < -2 or x > 2
Steps:
x²/4 > 1
x² > 4
x < -2 or x > 2
12. Which of the following is the solution to |x + 1| + |x - 1| < 4?
A) -1 < x < 1
B) -2 < x < 2
C) -3 < x < 3
D) -4 < x < 4
Answer: B) -2 < x < 2
Steps:
For x ≥ 1: x + 1 + x - 1 < 4, 2x < 4, x < 2
For -1 ≤ x < 1: x + 1 - x + 1 < 4, 2 < 4 (always true)
For x < -1: -x - 1 - x + 1 < 4, -2x < 4, x > -2
Combining these conditions: -2 < x < 2
13. Solve: (x - 2)(x + 3) ≤ 0
A) -3 ≤ x ≤ 2
B) x ≤ -3 or x ≥ 2
C) -3 < x < 2
D) x < -3 or x > 2
Answer: A) -3 ≤ x ≤ 2
Steps:
The product is non-positive when one factor is non-positive and the other is non-negative.
x - 2 ≤ 0 and x + 3 ≥ 0, or
x ≤ 2 and x ≥ -3
Therefore, -3 ≤ x ≤ 2
14. What is the solution set of 2x + 1 < 7 or 3x - 2 ≥ 10?
A) x < 3 or x ≥ 4
B) x > 3 and x < 4
C) x ≤ 3 or x > 4
D) x < 3 and x > 4
Answer: A) x < 3 or x ≥ 4
Steps:
2x + 1 < 7 → x < 3
3x - 2 ≥ 10 → x ≥ 4
Combine: x < 3 or x ≥ 4
15. Solve: √(x + 4) < 3
A) -4 < x < 5
B) -4 ≤ x < 5
C) -4 < x ≤ 5
D) -4 ≤ x ≤ 5
Answer: B) -4 ≤ x < 5
Steps:
√(x + 4) < 3
x + 4 < 9
x < 5
Also, x + 4 ≥ 0 (for real solutions)
x ≥ -4
Combining: -4 ≤ x < 5
16. Which inequality is represented by the graph?
[Image of a number line with an open circle at 2 and shading to the right]
A) x > 2
B) x < 2
C) x ≥ 2
D) x ≤ 2
Answer: A) x > 2
17. Solve: 3(2x - 1) < 4x + 5
A) x > 4
B) x < 4
C) x > -4
D) x < -4
Answer: B) x < 4
Steps:
6x - 3 < 4x + 5
2x < 8
x < 4
18. What is the solution set of |3x - 2| ≥ 4?
A) x ≤ -2/3 or x ≥ 2
B) -2/3 < x < 2
C) x < -2/3 or x > 2
D) x ≤ 2/3 or x ≥ 2
Answer: C) x < -2/3 or x > 2
Steps:
|3x - 2| ≥ 4
3x - 2 ≤ -4 or 3x - 2 ≥ 4
3x ≤ -2 or 3x ≥ 6
x ≤ -2/3 or x ≥ 2
19. Solve: x² + 6x + 8 > 0
A) x < -4 or x > -2
B) -4 < x < -2
C) x ≤ -4 or x ≥ -2
D) -4 ≤ x ≤ -2
Answer: A) x < -4 or x > -2
Steps:
Factoring: (x + 4)(x + 2) > 0
The inequality is satisfied when both factors are positive or both are negative.
x < -4 or x > -2
20. Which of the following is equivalent to 2 < x ≤ 5?
A) x > 2 and x < 5
B) x > 2 and x ≤ 5
C) x ≥ 2 and x ≤ 5
D) x > 2 or x ≤ 5
Answer: B) x > 2 and x ≤ 5
21. Solve: 2x - 3 > 7 or 3x + 1 ≤ 4
A) x > 5 or x ≤ 1
B) x < 5 and x > 1
C) x ≥ 5 or x < 1
D) x > 5 and x < 1
Answer: A) x > 5 or x ≤ 1
Steps:
2x - 3 > 7 → x > 5
3x + 1 ≤ 4 → x ≤ 1
Combine: x > 5 or x ≤ 1
22. What is the solution set of x² - 5x + 6 ≤ 0?
A) 2 ≤ x ≤ 3
B) x < 2 or x > 3
C) 2 < x < 3
D) x ≤ 2 or x ≥ 3
Answer: A) 2 ≤ x ≤ 3
Steps:
Factoring: (x - 2)(x - 3) ≤ 0
The inequality is satisfied when one factor is non-positive and the other is non-negative.
2 ≤ x ≤ 3
23. Solve: 1/x < 1/2
A) x > 2
B) 0 < x < 2
C) x < 0 or x > 2
D) x < 2, x ≠ 0
Answer: C) x < 0 or x > 2
Steps:
1/x < 1/2
2 < x (for x > 0)
2 > x (for x < 0)
Therefore, x < 0 or x > 2
24. Which inequality represents the shaded region?
[Image of a number line with shading to the right of 3, including 3]
A) x > 3
B) x < 3
C) x ≥ 3
D) x ≤ 3
Answer: C) x ≥ 3
25. Solve: |2x + 1| - 3 < 0
A) -2 < x < 1
B) -2 ≤ x ≤ 1
C) -1 < x < 2
D) -1 ≤ x ≤ 2
Answer: A) -2 < x < 1
Steps:
|2x + 1| < 3
-3 < 2x + 1 < 3
-4 < 2x < 2
-2 < x < 1
26. Which of the following is the solution to 3x - 7 ≥ 2x + 5?
A) x ≥ 12
B) x ≤ 12
C) x ≥ -12
D) x ≤ -12
Answer: A) x ≥ 12
Steps:
3x - 7 ≥ 2x + 5
x - 7 ≥ 5
x ≥ 12
27. Solve: x²/9 + y²/4 ≤ 1
A) Ellipse with center (0,0), a = 3, b = 2
B) Ellipse with center (0,0), a = 2, b = 3
C) Circle with center (0,0), radius 1
D) Hyperbola with center (0,0)
Answer: A) Ellipse with center (0,0), a = 3, b = 2
Steps:
This is the standard form of an ellipse equation where a² = 9 and b² = 4.
Therefore, a = 3 and b = 2.
28. What is the solution set of |x - 1| + |x + 1| ≤ 4?
A) -1 ≤ x ≤ 1
B) -2 ≤ x ≤ 2
C) -3 ≤ x ≤ 3
D) -4 ≤ x ≤ 4
Answer: B) -2 ≤ x ≤ 2
Steps:
For x ≥ 1: x - 1 + x + 1 ≤ 4, 2x ≤ 4, x ≤ 2
For -1 ≤ x < 1: -x + 1 + x + 1 ≤ 4, 2 ≤ 4 (always true)
For x < -1: -x + 1 - x - 1 ≤ 4, -2x ≤ 4, x
Solve: x² - 4x - 5 < 0
A) -1 < x < 5
B) x < -1 or x > 5
C) -1 < x < 5
D) x < -1 and x > 5
Answer: A) -1 < x < 5
Steps:
Factoring: (x - 5)(x + 1) < 0
The inequality is satisfied when one factor is negative and the other is positive.
x - 5 < 0 and x + 1 > 0, or
x < 5 and x > -1
Therefore, -1 < x < 5
6. Which of the following is equivalent to x > 2 and x < 5?
A) 2 < x < 5
B) x > 2 or x < 5
C) x ≥ 2 and x ≤ 5
D) 2 ≤ x < 5
Answer: A) 2 < x < 5
7. Solve: 2(x - 3) ≥ 4x + 2
A) x ≤ -4
B) x ≥ -4
C) x ≤ 4
D) x ≥ 4
Answer: A) x ≤ -4
Steps:
2x - 6 ≥ 4x + 2
-2x ≥ 8
x ≤ -4
8. What is the solution set of |2x - 1| > 5?
A) x < -2 or x > 3
B) -2 < x < 3
C) x < 3 or x > -2
D) x < -2 and x > 3
Answer: A) x < -2 or x > 3
Steps:
|2x - 1| > 5
2x - 1 < -5 or 2x - 1 > 5
2x < -4 or 2x > 6
x < -2 or x > 3
9. Solve: 3x + 2 < x - 4
A) x > 3
B) x < -3
C) x > -3
D) x < 3
Answer: B) x < -3
Steps:
3x + 2 < x - 4
2x < -6
x < -3
10. Which inequality represents the solution to 5 - 2x ≤ 1?
A) x ≥ 2
B) x ≤ 2
C) x ≥ -2
D) x ≤ -2
Answer: A) x ≥ 2
Steps:
5 - 2x ≤ 1
-2x ≤ -4
x ≥ 2
11. Solve: x²/4 - 1 > 0
A) x < -2 or x > 2
B) -2 < x < 2
C) x ≤ -2 or x ≥ 2
D) -2 ≤ x ≤ 2
Answer: A) x < -2 or x > 2
Steps:
x²/4 > 1
x² > 4
x < -2 or x > 2
12. Which of the following is the solution to |x + 1| + |x - 1| < 4?
A) -1 < x < 1
B) -2 < x < 2
C) -3 < x < 3
D) -4 < x < 4
Answer: B) -2 < x < 2
Steps:
For x ≥ 1: x + 1 + x - 1 < 4, 2x < 4, x < 2
For -1 ≤ x < 1: x + 1 - x + 1 < 4, 2 < 4 (always true)
For x < -1: -x - 1 - x + 1 < 4, -2x < 4, x > -2
Combining these conditions: -2 < x < 2
13. Solve: (x - 2)(x + 3) ≤ 0
A) -3 ≤ x ≤ 2
B) x ≤ -3 or x ≥ 2
C) -3 < x < 2
D) x < -3 or x > 2
Answer: A) -3 ≤ x ≤ 2
Steps:
The product is non-positive when one factor is non-positive and the other is non-negative.
x - 2 ≤ 0 and x + 3 ≥ 0, or
x ≤ 2 and x ≥ -3
Therefore, -3 ≤ x ≤ 2
14. What is the solution set of 2x + 1 < 7 or 3x - 2 ≥ 10?
A) x < 3 or x ≥ 4
B) x > 3 and x < 4
C) x ≤ 3 or x > 4
D) x < 3 and x > 4
Answer: A) x < 3 or x ≥ 4
Steps:
2x + 1 < 7 → x < 3
3x - 2 ≥ 10 → x ≥ 4
Combine: x < 3 or x ≥ 4
15. Solve: √(x + 4) < 3
A) -4 < x < 5
B) -4 ≤ x < 5
C) -4 < x ≤ 5
D) -4 ≤ x ≤ 5
Answer: B) -4 ≤ x < 5
Steps:
√(x + 4) < 3
x + 4 < 9
x < 5
Also, x + 4 ≥ 0 (for real solutions)
x ≥ -4
Combining: -4 ≤ x < 5
16. Which inequality is represented by the graph?
[Image of a number line with an open circle at 2 and shading to the right]
A) x > 2
B) x < 2
C) x ≥ 2
D) x ≤ 2
Answer: A) x > 2
17. Solve: 3(2x - 1) < 4x + 5
A) x > 4
B) x < 4
C) x > -4
D) x < -4
Answer: B) x < 4
Steps:
6x - 3 < 4x + 5
2x < 8
x < 4
18. What is the solution set of |3x - 2| ≥ 4?
A) x ≤ -2/3 or x ≥ 2
B) -2/3 < x < 2
C) x < -2/3 or x > 2
D) x ≤ 2/3 or x ≥ 2
Answer: C) x < -2/3 or x > 2
Steps:
|3x - 2| ≥ 4
3x - 2 ≤ -4 or 3x - 2 ≥ 4
3x ≤ -2 or 3x ≥ 6
x ≤ -2/3 or x ≥ 2
19. Solve: x² + 6x + 8 > 0
A) x < -4 or x > -2
B) -4 < x < -2
C) x ≤ -4 or x ≥ -2
D) -4 ≤ x ≤ -2
Answer: A) x < -4 or x > -2
Steps:
Factoring: (x + 4)(x + 2) > 0
The inequality is satisfied when both factors are positive or both are negative.
x < -4 or x > -2
20. Which of the following is equivalent to 2 < x ≤ 5?
A) x > 2 and x < 5
B) x > 2 and x ≤ 5
C) x ≥ 2 and x ≤ 5
D) x > 2 or x ≤ 5
Answer: B) x > 2 and x ≤ 5
21. Solve: 2x - 3 > 7 or 3x + 1 ≤ 4
A) x > 5 or x ≤ 1
B) x < 5 and x > 1
C) x ≥ 5 or x < 1
D) x > 5 and x < 1
Answer: A) x > 5 or x ≤ 1
Steps:
2x - 3 > 7 → x > 5
3x + 1 ≤ 4 → x ≤ 1
Combine: x > 5 or x ≤ 1
22. What is the solution set of x² - 5x + 6 ≤ 0?
A) 2 ≤ x ≤ 3
B) x < 2 or x > 3
C) 2 < x < 3
D) x ≤ 2 or x ≥ 3
Answer: A) 2 ≤ x ≤ 3
Steps:
Factoring: (x - 2)(x - 3) ≤ 0
The inequality is satisfied when one factor is non-positive and the other is non-negative.
2 ≤ x ≤ 3
23. Solve: 1/x < 1/2
A) x > 2
B) 0 < x < 2
C) x < 0 or x > 2
D) x < 2, x ≠ 0
Answer: C) x < 0 or x > 2
Steps:
1/x < 1/2
2 < x (for x > 0)
2 > x (for x < 0)
Therefore, x < 0 or x > 2
24. Which inequality represents the shaded region?
[Image of a number line with shading to the right of 3, including 3]
A) x > 3
B) x < 3
C) x ≥ 3
D) x ≤ 3
Answer: C) x ≥ 3
25. Solve: |2x + 1| - 3 < 0
A) -2 < x < 1
B) -2 ≤ x ≤ 1
C) -1 < x < 2
D) -1 ≤ x ≤ 2
Answer: A) -2 < x < 1
Steps:
|2x + 1| < 3
-3 < 2x + 1 < 3
-4 < 2x < 2
-2 < x < 1
26. Which of the following is the solution to 3x - 7 ≥ 2x + 5?
A) x ≥ 12
B) x ≤ 12
C) x ≥ -12
D) x ≤ -12
Answer: A) x ≥ 12
Steps:
3x - 7 ≥ 2x + 5
x - 7 ≥ 5
x ≥ 12
27. Solve: x²/9 + y²/4 ≤ 1
A) Ellipse with center (0,0), a = 3, b = 2
B) Ellipse with center (0,0), a = 2, b = 3
C) Circle with center (0,0), radius 1
D) Hyperbola with center (0,0)
Answer: A) Ellipse with center (0,0), a = 3, b = 2
Steps:
This is the standard form of an ellipse equation where a² = 9 and b² = 4.
Therefore, a = 3 and b = 2.
28. What is the solution set of |x - 1| + |x + 1| ≤ 4?
A) -1 ≤ x ≤ 1
B) -2 ≤ x ≤ 2
C) -3 ≤ x ≤ 3
D) -4 ≤ x ≤ 4
Answer: B) -2 ≤ x ≤ 2
Steps:
For x ≥ 1: x - 1 + x + 1 ≤ 4, 2x ≤ 4, x ≤ 2
For -1 ≤ x < 1: -x + 1 + x + 1 ≤ 4, 2 ≤ 4 (always true)
For x < -1: -x + 1 - x - 1 ≤ 4, -2x ≤ 4, x
Solve: x² - 4x - 5 < 0
A) -1 < x < 5
B) x < -1 or x > 5
C) -1 < x < 5
D) x < -1 and x > 5
Answer: A) -1 < x < 5
Steps:
Factoring: (x - 5)(x + 1) < 0
The inequality is satisfied when one factor is negative and the other is positive.
x - 5 < 0 and x + 1 > 0, or
x < 5 and x > -1
Therefore, -1 < x < 5
6. Which of the following is equivalent to x > 2 and x < 5?
A) 2 < x < 5
B) x > 2 or x < 5
C) x ≥ 2 and x ≤ 5
D) 2 ≤ x < 5
Answer: A) 2 < x < 5
7. Solve: 2(x - 3) ≥ 4x + 2
A) x ≤ -4
B) x ≥ -4
C) x ≤ 4
D) x ≥ 4
Answer: A) x ≤ -4
Steps:
2x - 6 ≥ 4x + 2
-2x ≥ 8
x ≤ -4
8. What is the solution set of |2x - 1| > 5?
A) x < -2 or x > 3
B) -2 < x < 3
C) x < 3 or x > -2
D) x < -2 and x > 3
Answer: A) x < -2 or x > 3
Steps:
|2x - 1| > 5
2x - 1 < -5 or 2x - 1 > 5
2x < -4 or 2x > 6
x < -2 or x > 3
9. Solve: 3x + 2 < x - 4
A) x > 3
B) x < -3
C) x > -3
D) x < 3
Answer: B) x < -3
Steps:
3x + 2 < x - 4
2x < -6
x < -3
10. Which inequality represents the solution to 5 - 2x ≤ 1?
A) x ≥ 2
B) x ≤ 2
C) x ≥ -2
D) x ≤ -2
Answer: A) x ≥ 2
Steps:
5 - 2x ≤ 1
-2x ≤ -4
x ≥ 2
11. Solve: x²/4 - 1 > 0
A) x < -2 or x > 2
B) -2 < x < 2
C) x ≤ -2 or x ≥ 2
D) -2 ≤ x ≤ 2
Answer: A) x < -2 or x > 2
Steps:
x²/4 > 1
x² > 4
x < -2 or x > 2
12. Which of the following is the solution to |x + 1| + |x - 1| < 4?
A) -1 < x < 1
B) -2 < x < 2
C) -3 < x < 3
D) -4 < x < 4
Answer: B) -2 < x < 2
Steps:
For x ≥ 1: x + 1 + x - 1 < 4, 2x < 4, x < 2
For -1 ≤ x < 1: x + 1 - x + 1 < 4, 2 < 4 (always true)
For x < -1: -x - 1 - x + 1 < 4, -2x < 4, x > -2
Combining these conditions: -2 < x < 2
13. Solve: (x - 2)(x + 3) ≤ 0
A) -3 ≤ x ≤ 2
B) x ≤ -3 or x ≥ 2
C) -3 < x < 2
D) x < -3 or x > 2
Answer: A) -3 ≤ x ≤ 2
Steps:
The product is non-positive when one factor is non-positive and the other is non-negative.
x - 2 ≤ 0 and x + 3 ≥ 0, or
x ≤ 2 and x ≥ -3
Therefore, -3 ≤ x ≤ 2
14. What is the solution set of 2x + 1 < 7 or 3x - 2 ≥ 10?
A) x < 3 or x ≥ 4
B) x > 3 and x < 4
C) x ≤ 3 or x > 4
D) x < 3 and x > 4
Answer: A) x < 3 or x ≥ 4
Steps:
2x + 1 < 7 → x < 3
3x - 2 ≥ 10 → x ≥ 4
Combine: x < 3 or x ≥ 4
15. Solve: √(x + 4) < 3
A) -4 < x < 5
B) -4 ≤ x < 5
C) -4 < x ≤ 5
D) -4 ≤ x ≤ 5
Answer: B) -4 ≤ x < 5
Steps:
√(x + 4) < 3
x + 4 < 9
x < 5
Also, x + 4 ≥ 0 (for real solutions)
x ≥ -4
Combining: -4 ≤ x < 5
16. Which inequality is represented by the graph?
[Image of a number line with an open circle at 2 and shading to the right]
A) x > 2
B) x < 2
C) x ≥ 2
D) x ≤ 2
Answer: A) x > 2
17. Solve: 3(2x - 1) < 4x + 5
A) x > 4
B) x < 4
C) x > -4
D) x < -4
Answer: B) x < 4
Steps:
6x - 3 < 4x + 5
2x < 8
x < 4
18. What is the solution set of |3x - 2| ≥ 4?
A) x ≤ -2/3 or x ≥ 2
B) -2/3 < x < 2
C) x < -2/3 or x > 2
D) x ≤ 2/3 or x ≥ 2
Answer: C) x < -2/3 or x > 2
Steps:
|3x - 2| ≥ 4
3x - 2 ≤ -4 or 3x - 2 ≥ 4
3x ≤ -2 or 3x ≥ 6
x ≤ -2/3 or x ≥ 2
19. Solve: x² + 6x + 8 > 0
A) x < -4 or x > -2
B) -4 < x < -2
C) x ≤ -4 or x ≥ -2
D) -4 ≤ x ≤ -2
Answer: A) x < -4 or x > -2
Steps:
Factoring: (x + 4)(x + 2) > 0
The inequality is satisfied when both factors are positive or both are negative.
x < -4 or x > -2
20. Which of the following is equivalent to 2 < x ≤ 5?
A) x > 2 and x < 5
B) x > 2 and x ≤ 5
C) x ≥ 2 and x ≤ 5
D) x > 2 or x ≤ 5
Answer: B) x > 2 and x ≤ 5
21. Solve: 2x - 3 > 7 or 3x + 1 ≤ 4
A) x > 5 or x ≤ 1
B) x < 5 and x > 1
C) x ≥ 5 or x < 1
D) x > 5 and x < 1
Answer: A) x > 5 or x ≤ 1
Steps:
2x - 3 > 7 → x > 5
3x + 1 ≤ 4 → x ≤ 1
Combine: x > 5 or x ≤ 1
22. What is the solution set of x² - 5x + 6 ≤ 0?
A) 2 ≤ x ≤ 3
B) x < 2 or x > 3
C) 2 < x < 3
D) x ≤ 2 or x ≥ 3
Answer: A) 2 ≤ x ≤ 3
Steps:
Factoring: (x - 2)(x - 3) ≤ 0
The inequality is satisfied when one factor is non-positive and the other is non-negative.
2 ≤ x ≤ 3
23. Solve: 1/x < 1/2
A) x > 2
B) 0 < x < 2
C) x < 0 or x > 2
D) x < 2, x ≠ 0
Answer: C) x < 0 or x > 2
Steps:
1/x < 1/2
2 < x (for x > 0)
2 > x (for x < 0)
Therefore, x < 0 or x > 2
24. Which inequality represents the shaded region?
[Image of a number line with shading to the right of 3, including 3]
A) x > 3
B) x < 3
C) x ≥ 3
D) x ≤ 3
Answer: C) x ≥ 3
25. Solve: |2x + 1| - 3 < 0
A) -2 < x < 1
B) -2 ≤ x ≤ 1
C) -1 < x < 2
D) -1 ≤ x ≤ 2
Answer: A) -2 < x < 1
Steps:
|2x + 1| < 3
-3 < 2x + 1 < 3
-4 < 2x < 2
-2 < x < 1
26. Which of the following is the solution to 3x - 7 ≥ 2x + 5?
A) x ≥ 12
B) x ≤ 12
C) x ≥ -12
D) x ≤ -12
Answer: A) x ≥ 12
Steps:
3x - 7 ≥ 2x + 5
x - 7 ≥ 5
x ≥ 12
27. Solve: x²/9 + y²/4 ≤ 1
A) Ellipse with center (0,0), a = 3, b = 2
B) Ellipse with center (0,0), a = 2, b = 3
C) Circle with center (0,0), radius 1
D) Hyperbola with center (0,0)
Answer: A) Ellipse with center (0,0), a = 3, b = 2
Steps:
This is the standard form of an ellipse equation where a² = 9 and b² = 4.
Therefore, a = 3 and b = 2.
28. What is the solution set of |x - 1| + |x + 1| ≤ 4?
A) -1 ≤ x ≤ 1
B) -2 ≤ x ≤ 2
C) -3 ≤ x ≤ 3
D) -4 ≤ x ≤ 4
Answer: B) -2 ≤ x ≤ 2
Steps:
For x ≥ 1: x - 1 + x + 1 ≤ 4, 2x ≤ 4, x ≤ 2
For -1 ≤ x < 1: -x + 1 + x + 1 ≤ 4, 2 ≤ 4 (always true)
For x < -1: -x + 1 - x - 1 ≤ 4, -2x ≤ 4, x
Solve: x² - 4x - 5 < 0
A) -1 < x < 5
B) x < -1 or x > 5
C) -1 < x < 5
D) x < -1 and x > 5
Answer: A) -1 < x < 5
Steps:
Factoring: (x - 5)(x + 1) < 0
The inequality is satisfied when one factor is negative and the other is positive.
x - 5 < 0 and x + 1 > 0, or
x < 5 and x > -1
Therefore, -1 < x < 5
6. Which of the following is equivalent to x > 2 and x < 5?
A) 2 < x < 5
B) x > 2 or x < 5
C) x ≥ 2 and x ≤ 5
D) 2 ≤ x < 5
Answer: A) 2 < x < 5
7. Solve: 2(x - 3) ≥ 4x + 2
A) x ≤ -4
B) x ≥ -4
C) x ≤ 4
D) x ≥ 4
Answer: A) x ≤ -4
Steps:
2x - 6 ≥ 4x + 2
-2x ≥ 8
x ≤ -4
8. What is the solution set of |2x - 1| > 5?
A) x < -2 or x > 3
B) -2 < x < 3
C) x < 3 or x > -2
D) x < -2 and x > 3
Answer: A) x < -2 or x > 3
Steps:
|2x - 1| > 5
2x - 1 < -5 or 2x - 1 > 5
2x < -4 or 2x > 6
x < -2 or x > 3
9. Solve: 3x + 2 < x - 4
A) x > 3
B) x < -3
C) x > -3
D) x < 3
Answer: B) x < -3
Steps:
3x + 2 < x - 4
2x < -6
x < -3
10. Which inequality represents the solution to 5 - 2x ≤ 1?
A) x ≥ 2
B) x ≤ 2
C) x ≥ -2
D) x ≤ -2
Answer: A) x ≥ 2
Steps:
5 - 2x ≤ 1
-2x ≤ -4
x ≥ 2
11. Solve: x²/4 - 1 > 0
A) x < -2 or x > 2
B) -2 < x < 2
C) x ≤ -2 or x ≥ 2
D) -2 ≤ x ≤ 2
Answer: A) x < -2 or x > 2
Steps:
x²/4 > 1
x² > 4
x < -2 or x > 2
12. Which of the following is the solution to |x + 1| + |x - 1| < 4?
A) -1 < x < 1
B) -2 < x < 2
C) -3 < x < 3
D) -4 < x < 4
Answer: B) -2 < x < 2
Steps:
For x ≥ 1: x + 1 + x - 1 < 4, 2x < 4, x < 2
For -1 ≤ x < 1: x + 1 - x + 1 < 4, 2 < 4 (always true)
For x < -1: -x - 1 - x + 1 < 4, -2x < 4, x > -2
Combining these conditions: -2 < x < 2
13. Solve: (x - 2)(x + 3) ≤ 0
A) -3 ≤ x ≤ 2
B) x ≤ -3 or x ≥ 2
C) -3 < x < 2
D) x < -3 or x > 2
Answer: A) -3 ≤ x ≤ 2
Steps:
The product is non-positive when one factor is non-positive and the other is non-negative.
x - 2 ≤ 0 and x + 3 ≥ 0, or
x ≤ 2 and x ≥ -3
Therefore, -3 ≤ x ≤ 2
14. What is the solution set of 2x + 1 < 7 or 3x - 2 ≥ 10?
A) x < 3 or x ≥ 4
B) x > 3 and x < 4
C) x ≤ 3 or x > 4
D) x < 3 and x > 4
Answer: A) x < 3 or x ≥ 4
Steps:
2x + 1 < 7 → x < 3
3x - 2 ≥ 10 → x ≥ 4
Combine: x < 3 or x ≥ 4
15. Solve: √(x + 4) < 3
A) -4 < x < 5
B) -4 ≤ x < 5
C) -4 < x ≤ 5
D) -4 ≤ x ≤ 5
Answer: B) -4 ≤ x < 5
Steps:
√(x + 4) < 3
x + 4 < 9
x < 5
Also, x + 4 ≥ 0 (for real solutions)
x ≥ -4
Combining: -4 ≤ x < 5
16. Which inequality is represented by the graph?
[Image of a number line with an open circle at 2 and shading to the right]
A) x > 2
B) x < 2
C) x ≥ 2
D) x ≤ 2
Answer: A) x > 2
17. Solve: 3(2x - 1) < 4x + 5
A) x > 4
B) x < 4
C) x > -4
D) x < -4
Answer: B) x < 4
Steps:
6x - 3 < 4x + 5
2x < 8
x < 4
18. What is the solution set of |3x - 2| ≥ 4?
A) x ≤ -2/3 or x ≥ 2
B) -2/3 < x < 2
C) x < -2/3 or x > 2
D) x ≤ 2/3 or x ≥ 2
Answer: C) x < -2/3 or x > 2
Steps:
|3x - 2| ≥ 4
3x - 2 ≤ -4 or 3x - 2 ≥ 4
3x ≤ -2 or 3x ≥ 6
x ≤ -2/3 or x ≥ 2
19. Solve: x² + 6x + 8 > 0
A) x < -4 or x > -2
B) -4 < x < -2
C) x ≤ -4 or x ≥ -2
D) -4 ≤ x ≤ -2
Answer: A) x < -4 or x > -2
Steps:
Factoring: (x + 4)(x + 2) > 0
The inequality is satisfied when both factors are positive or both are negative.
x < -4 or x > -2
20. Which of the following is equivalent to 2 < x ≤ 5?
A) x > 2 and x < 5
B) x > 2 and x ≤ 5
C) x ≥ 2 and x ≤ 5
D) x > 2 or x ≤ 5
Answer: B) x > 2 and x ≤ 5
21. Solve: 2x - 3 > 7 or 3x + 1 ≤ 4
A) x > 5 or x ≤ 1
B) x < 5 and x > 1
C) x ≥ 5 or x < 1
D) x > 5 and x < 1
Answer: A) x > 5 or x ≤ 1
Steps:
2x - 3 > 7 → x > 5
3x + 1 ≤ 4 → x ≤ 1
Combine: x > 5 or x ≤ 1
22. What is the solution set of x² - 5x + 6 ≤ 0?
A) 2 ≤ x ≤ 3
B) x < 2 or x > 3
C) 2 < x < 3
D) x ≤ 2 or x ≥ 3
Answer: A) 2 ≤ x ≤ 3
Steps:
Factoring: (x - 2)(x - 3) ≤ 0
The inequality is satisfied when one factor is non-positive and the other is non-negative.
2 ≤ x ≤ 3
23. Solve: 1/x < 1/2
A) x > 2
B) 0 < x < 2
C) x < 0 or x > 2
D) x < 2, x ≠ 0
Answer: C) x < 0 or x > 2
Steps:
1/x < 1/2
2 < x (for x > 0)
2 > x (for x < 0)
Therefore, x < 0 or x > 2
24. Which inequality represents the shaded region?
[Image of a number line with shading to the right of 3, including 3]
A) x > 3
B) x < 3
C) x ≥ 3
D) x ≤ 3
Answer: C) x ≥ 3
25. Solve: |2x + 1| - 3 < 0
A) -2 < x < 1
B) -2 ≤ x ≤ 1
C) -1 < x < 2
D) -1 ≤ x ≤ 2
Answer: A) -2 < x < 1
Steps:
|2x + 1| < 3
-3 < 2x + 1 < 3
-4 < 2x < 2
-2 < x < 1
26. Which of the following is the solution to 3x - 7 ≥ 2x + 5?
A) x ≥ 12
B) x ≤ 12
C) x ≥ -12
D) x ≤ -12
Answer: A) x ≥ 12
Steps:
3x - 7 ≥ 2x + 5
x - 7 ≥ 5
x ≥ 12
27. Solve: x²/9 + y²/4 ≤ 1
A) Ellipse with center (0,0), a = 3, b = 2
B) Ellipse with center (0,0), a = 2, b = 3
C) Circle with center (0,0), radius 1
D) Hyperbola with center (0,0)
Answer: A) Ellipse with center (0,0), a = 3, b = 2
Steps:
This is the standard form of an ellipse equation where a² = 9 and b² = 4.
Therefore, a = 3 and b = 2.
28. What is the solution set of |x - 1| + |x + 1| ≤ 4?
A) -1 ≤ x ≤ 1
B) -2 ≤ x ≤ 2
C) -3 ≤ x ≤ 3
D) -4 ≤ x ≤ 4
Answer: B) -2 ≤ x ≤ 2
Steps:
For x ≥ 1: x - 1 + x + 1 ≤ 4, 2x ≤ 4, x ≤ 2
For -1 ≤ x < 1: -x + 1 + x + 1 ≤ 4, 2 ≤ 4 (always true)
For x < -1: -x + 1 - x - 1 ≤ 4, -2x ≤ 4, x
Solve: x² - 4x - 5 < 0
A) -1 < x < 5
B) x < -1 or x > 5
C) -1 < x < 5
D) x < -1 and x > 5
Answer: A) -1 < x < 5
Steps:
Factoring: (x - 5)(x + 1) < 0
The inequality is satisfied when one factor is negative and the other is positive.
x - 5 < 0 and x + 1 > 0, or
x < 5 and x > -1
Therefore, -1 < x < 5
6. Which of the following is equivalent to x > 2 and x < 5?
A) 2 < x < 5
B) x > 2 or x < 5
C) x ≥ 2 and x ≤ 5
D) 2 ≤ x < 5
Answer: A) 2 < x < 5
7. Solve: 2(x - 3) ≥ 4x + 2
A) x ≤ -4
B) x ≥ -4
C) x ≤ 4
D) x ≥ 4
Answer: A) x ≤ -4
Steps:
2x - 6 ≥ 4x + 2
-2x ≥ 8
x ≤ -4
8. What is the solution set of |2x - 1| > 5?
A) x < -2 or x > 3
B) -2 < x < 3
C) x < 3 or x > -2
D) x < -2 and x > 3
Answer: A) x < -2 or x > 3
Steps:
|2x - 1| > 5
2x - 1 < -5 or 2x - 1 > 5
2x < -4 or 2x > 6
x < -2 or x > 3
9. Solve: 3x + 2 < x - 4
A) x > 3
B) x < -3
C) x > -3
D) x < 3
Answer: B) x < -3
Steps:
3x + 2 < x - 4
2x < -6
x < -3
10. Which inequality represents the solution to 5 - 2x ≤ 1?
A) x ≥ 2
B) x ≤ 2
C) x ≥ -2
D) x ≤ -2
Answer: A) x ≥ 2
Steps:
5 - 2x ≤ 1
-2x ≤ -4
x ≥ 2
11. Solve: x²/4 - 1 > 0
A) x < -2 or x > 2
B) -2 < x < 2
C) x ≤ -2 or x ≥ 2
D) -2 ≤ x ≤ 2
Answer: A) x < -2 or x > 2
Steps:
x²/4 > 1
x² > 4
x < -2 or x > 2
12. Which of the following is the solution to |x + 1| + |x - 1| < 4?
A) -1 < x < 1
B) -2 < x < 2
C) -3 < x < 3
D) -4 < x < 4
Answer: B) -2 < x < 2
Steps:
For x ≥ 1: x + 1 + x - 1 < 4, 2x < 4, x < 2
For -1 ≤ x < 1: x + 1 - x + 1 < 4, 2 < 4 (always true)
For x < -1: -x - 1 - x + 1 < 4, -2x < 4, x > -2
Combining these conditions: -2 < x < 2
13. Solve: (x - 2)(x + 3) ≤ 0
A) -3 ≤ x ≤ 2
B) x ≤ -3 or x ≥ 2
C) -3 < x < 2
D) x < -3 or x > 2
Answer: A) -3 ≤ x ≤ 2
Steps:
The product is non-positive when one factor is non-positive and the other is non-negative.
x - 2 ≤ 0 and x + 3 ≥ 0, or
x ≤ 2 and x ≥ -3
Therefore, -3 ≤ x ≤ 2
14. What is the solution set of 2x + 1 < 7 or 3x - 2 ≥ 10?
A) x < 3 or x ≥ 4
B) x > 3 and x < 4
C) x ≤ 3 or x > 4
D) x < 3 and x > 4
Answer: A) x < 3 or x ≥ 4
Steps:
2x + 1 < 7 → x < 3
3x - 2 ≥ 10 → x ≥ 4
Combine: x < 3 or x ≥ 4
15. Solve: √(x + 4) < 3
A) -4 < x < 5
B) -4 ≤ x < 5
C) -4 < x ≤ 5
D) -4 ≤ x ≤ 5
Answer: B) -4 ≤ x < 5
Steps:
√(x + 4) < 3
x + 4 < 9
x < 5
Also, x + 4 ≥ 0 (for real solutions)
x ≥ -4
Combining: -4 ≤ x < 5
16. Which inequality is represented by the graph?
[Image of a number line with an open circle at 2 and shading to the right]
A) x > 2
B) x < 2
C) x ≥ 2
D) x ≤ 2
Answer: A) x > 2
17. Solve: 3(2x - 1) < 4x + 5
A) x > 4
B) x < 4
C) x > -4
D) x < -4
Answer: B) x < 4
Steps:
6x - 3 < 4x + 5
2x < 8
x < 4
18. What is the solution set of |3x - 2| ≥ 4?
A) x ≤ -2/3 or x ≥ 2
B) -2/3 < x < 2
C) x < -2/3 or x > 2
D) x ≤ 2/3 or x ≥ 2
Answer: C) x < -2/3 or x > 2
Steps:
|3x - 2| ≥ 4
3x - 2 ≤ -4 or 3x - 2 ≥ 4
3x ≤ -2 or 3x ≥ 6
x ≤ -2/3 or x ≥ 2
19. Solve: x² + 6x + 8 > 0
A) x < -4 or x > -2
B) -4 < x < -2
C) x ≤ -4 or x ≥ -2
D) -4 ≤ x ≤ -2
Answer: A) x < -4 or x > -2
Steps:
Factoring: (x + 4)(x + 2) > 0
The inequality is satisfied when both factors are positive or both are negative.
x < -4 or x > -2
20. Which of the following is equivalent to 2 < x ≤ 5?
A) x > 2 and x < 5
B) x > 2 and x ≤ 5
C) x ≥ 2 and x ≤ 5
D) x > 2 or x ≤ 5
Answer: B) x > 2 and x ≤ 5
21. Solve: 2x - 3 > 7 or 3x + 1 ≤ 4
A) x > 5 or x ≤ 1
B) x < 5 and x > 1
C) x ≥ 5 or x < 1
D) x > 5 and x < 1
Answer: A) x > 5 or x ≤ 1
Steps:
2x - 3 > 7 → x > 5
3x + 1 ≤ 4 → x ≤ 1
Combine: x > 5 or x ≤ 1
22. What is the solution set of x² - 5x + 6 ≤ 0?
A) 2 ≤ x ≤ 3
B) x < 2 or x > 3
C) 2 < x < 3
D) x ≤ 2 or x ≥ 3
Answer: A) 2 ≤ x ≤ 3
Steps:
Factoring: (x - 2)(x - 3) ≤ 0
The inequality is satisfied when one factor is non-positive and the other is non-negative.
2 ≤ x ≤ 3
23. Solve: 1/x < 1/2
A) x > 2
B) 0 < x < 2
C) x < 0 or x > 2
D) x < 2, x ≠ 0
Answer: C) x < 0 or x > 2
Steps:
1/x < 1/2
2 < x (for x > 0)
2 > x (for x < 0)
Therefore, x < 0 or x > 2
24. Which inequality represents the shaded region?
[Image of a number line with shading to the right of 3, including 3]
A) x > 3
B) x < 3
C) x ≥ 3
D) x ≤ 3
Answer: C) x ≥ 3
25. Solve: |2x + 1| - 3 < 0
A) -2 < x < 1
B) -2 ≤ x ≤ 1
C) -1 < x < 2
D) -1 ≤ x ≤ 2
Answer: A) -2 < x < 1
Steps:
|2x + 1| < 3
-3 < 2x + 1 < 3
-4 < 2x < 2
-2 < x < 1
26. Which of the following is the solution to 3x - 7 ≥ 2x + 5?
A) x ≥ 12
B) x ≤ 12
C) x ≥ -12
D) x ≤ -12
Answer: A) x ≥ 12
Steps:
3x - 7 ≥ 2x + 5
x - 7 ≥ 5
x ≥ 12
27. Solve: x²/9 + y²/4 ≤ 1
A) Ellipse with center (0,0), a = 3, b = 2
B) Ellipse with center (0,0), a = 2, b = 3
C) Circle with center (0,0), radius 1
D) Hyperbola with center (0,0)
Answer: A) Ellipse with center (0,0), a = 3, b = 2
Steps:
This is the standard form of an ellipse equation where a² = 9 and b² = 4.
Therefore, a = 3 and b = 2.
28. What is the solution set of |x - 1| + |x + 1| ≤ 4?
A) -1 ≤ x ≤ 1
B) -2 ≤ x ≤ 2
C) -3 ≤ x ≤ 3
D) -4 ≤ x ≤ 4
Answer: B) -2 ≤ x ≤ 2
Steps:
For x ≥ 1: x - 1 + x + 1 ≤ 4, 2x ≤ 4, x ≤ 2
For -1 ≤ x < 1: -x + 1 + x + 1 ≤ 4, 2 ≤ 4 (always true)
For x < -1: -x + 1 - x - 1 ≤ 4, -2x ≤ 4, x
Solve: x² - 4x - 5 < 0
A) -1 < x < 5
B) x < -1 or x > 5
C) -1 < x < 5
D) x < -1 and x > 5
Answer: A) -1 < x < 5
Steps:
Factoring: (x - 5)(x + 1) < 0
The inequality is satisfied when one factor is negative and the other is positive.
x - 5 < 0 and x + 1 > 0, or
x < 5 and x > -1
Therefore, -1 < x < 5
6. Which of the following is equivalent to x > 2 and x < 5?
A) 2 < x < 5
B) x > 2 or x < 5
C) x ≥ 2 and x ≤ 5
D) 2 ≤ x < 5
Answer: A) 2 < x < 5
7. Solve: 2(x - 3) ≥ 4x + 2
A) x ≤ -4
B) x ≥ -4
C) x ≤ 4
D) x ≥ 4
Answer: A) x ≤ -4
Steps:
2x - 6 ≥ 4x + 2
-2x ≥ 8
x ≤ -4
8. What is the solution set of |2x - 1| > 5?
A) x < -2 or x > 3
B) -2 < x < 3
C) x < 3 or x > -2
D) x < -2 and x > 3
Answer: A) x < -2 or x > 3
Steps:
|2x - 1| > 5
2x - 1 < -5 or 2x - 1 > 5
2x < -4 or 2x > 6
x < -2 or x > 3
9. Solve: 3x + 2 < x - 4
A) x > 3
B) x < -3
C) x > -3
D) x < 3
Answer: B) x < -3
Steps:
3x + 2 < x - 4
2x < -6
x < -3
10. Which inequality represents the solution to 5 - 2x ≤ 1?
A) x ≥ 2
B) x ≤ 2
C) x ≥ -2
D) x ≤ -2
Answer: A) x ≥ 2
Steps:
5 - 2x ≤ 1
-2x ≤ -4
x ≥ 2
11. Solve: x²/4 - 1 > 0
A) x < -2 or x > 2
B) -2 < x < 2
C) x ≤ -2 or x ≥ 2
D) -2 ≤ x ≤ 2
Answer: A) x < -2 or x > 2
Steps:
x²/4 > 1
x² > 4
x < -2 or x > 2
12. Which of the following is the solution to |x + 1| + |x - 1| < 4?
A) -1 < x < 1
B) -2 < x < 2
C) -3 < x < 3
D) -4 < x < 4
Answer: B) -2 < x < 2
Steps:
For x ≥ 1: x + 1 + x - 1 < 4, 2x < 4, x < 2
For -1 ≤ x < 1: x + 1 - x + 1 < 4, 2 < 4 (always true)
For x < -1: -x - 1 - x + 1 < 4, -2x < 4, x > -2
Combining these conditions: -2 < x < 2
13. Solve: (x - 2)(x + 3) ≤ 0
A) -3 ≤ x ≤ 2
B) x ≤ -3 or x ≥ 2
C) -3 < x < 2
D) x < -3 or x > 2
Answer: A) -3 ≤ x ≤ 2
Steps:
The product is non-positive when one factor is non-positive and the other is non-negative.
x - 2 ≤ 0 and x + 3 ≥ 0, or
x ≤ 2 and x ≥ -3
Therefore, -3 ≤ x ≤ 2
14. What is the solution set of 2x + 1 < 7 or 3x - 2 ≥ 10?
A) x < 3 or x ≥ 4
B) x > 3 and x < 4
C) x ≤ 3 or x > 4
D) x < 3 and x > 4
Answer: A) x < 3 or x ≥ 4
Steps:
2x + 1 < 7 → x < 3
3x - 2 ≥ 10 → x ≥ 4
Combine: x < 3 or x ≥ 4
15. Solve: √(x + 4) < 3
A) -4 < x < 5
B) -4 ≤ x < 5
C) -4 < x ≤ 5
D) -4 ≤ x ≤ 5
Answer: B) -4 ≤ x < 5
Steps:
√(x + 4) < 3
x + 4 < 9
x < 5
Also, x + 4 ≥ 0 (for real solutions)
x ≥ -4
Combining: -4 ≤ x < 5
16. Which inequality is represented by the graph?
[Image of a number line with an open circle at 2 and shading to the right]
A) x > 2
B) x < 2
C) x ≥ 2
D) x ≤ 2
Answer: A) x > 2
17. Solve: 3(2x - 1) < 4x + 5
A) x > 4
B) x < 4
C) x > -4
D) x < -4
Answer: B) x < 4
Steps:
6x - 3 < 4x + 5
2x < 8
x < 4
18. What is the solution set of |3x - 2| ≥ 4?
A) x ≤ -2/3 or x ≥ 2
B) -2/3 < x < 2
C) x < -2/3 or x > 2
D) x ≤ 2/3 or x ≥ 2
Answer: C) x < -2/3 or x > 2
Steps:
|3x - 2| ≥ 4
3x - 2 ≤ -4 or 3x - 2 ≥ 4
3x ≤ -2 or 3x ≥ 6
x ≤ -2/3 or x ≥ 2
19. Solve: x² + 6x + 8 > 0
A) x < -4 or x > -2
B) -4 < x < -2
C) x ≤ -4 or x ≥ -2
D) -4 ≤ x ≤ -2
Answer: A) x < -4 or x > -2
Steps:
Factoring: (x + 4)(x + 2) > 0
The inequality is satisfied when both factors are positive or both are negative.
x < -4 or x > -2
20. Which of the following is equivalent to 2 < x ≤ 5?
A) x > 2 and x < 5
B) x > 2 and x ≤ 5
C) x ≥ 2 and x ≤ 5
D) x > 2 or x ≤ 5
Answer: B) x > 2 and x ≤ 5
21. Solve: 2x - 3 > 7 or 3x + 1 ≤ 4
A) x > 5 or x ≤ 1
B) x < 5 and x > 1
C) x ≥ 5 or x < 1
D) x > 5 and x < 1
Answer: A) x > 5 or x ≤ 1
Steps:
2x - 3 > 7 → x > 5
3x + 1 ≤ 4 → x ≤ 1
Combine: x > 5 or x ≤ 1
22. What is the solution set of x² - 5x + 6 ≤ 0?
A) 2 ≤ x ≤ 3
B) x < 2 or x > 3
C) 2 < x < 3
D) x ≤ 2 or x ≥ 3
Answer: A) 2 ≤ x ≤ 3
Steps:
Factoring: (x - 2)(x - 3) ≤ 0
The inequality is satisfied when one factor is non-positive and the other is non-negative.
2 ≤ x ≤ 3
23. Solve: 1/x < 1/2
A) x > 2
B) 0 < x < 2
C) x < 0 or x > 2
D) x < 2, x ≠ 0
Answer: C) x < 0 or x > 2
Steps:
1/x < 1/2
2 < x (for x > 0)
2 > x (for x < 0)
Therefore, x < 0 or x > 2
24. Which inequality represents the shaded region?
[Image of a number line with shading to the right of 3, including 3]
A) x > 3
B) x < 3
C) x ≥ 3
D) x ≤ 3
Answer: C) x ≥ 3
25. Solve: |2x + 1| - 3 < 0
A) -2 < x < 1
B) -2 ≤ x ≤ 1
C) -1 < x < 2
D) -1 ≤ x ≤ 2
Answer: A) -2 < x < 1
Steps:
|2x + 1| < 3
-3 < 2x + 1 < 3
-4 < 2x < 2
-2 < x < 1
26. Which of the following is the solution to 3x - 7 ≥ 2x + 5?
A) x ≥ 12
B) x ≤ 12
C) x ≥ -12
D) x ≤ -12
Answer: A) x ≥ 12
Steps:
3x - 7 ≥ 2x + 5
x - 7 ≥ 5
x ≥ 12
27. Solve: x²/9 + y²/4 ≤ 1
A) Ellipse with center (0,0), a = 3, b = 2
B) Ellipse with center (0,0), a = 2, b = 3
C) Circle with center (0,0), radius 1
D) Hyperbola with center (0,0)
Answer: A) Ellipse with center (0,0), a = 3, b = 2
Steps:
This is the standard form of an ellipse equation where a² = 9 and b² = 4.
Therefore, a = 3 and b = 2.
28. What is the solution set of |x - 1| + |x + 1| ≤ 4?
A) -1 ≤ x ≤ 1
B) -2 ≤ x ≤ 2
C) -3 ≤ x ≤ 3
D) -4 ≤ x ≤ 4
Answer: B) -2 ≤ x ≤ 2
Steps:
For x ≥ 1: x - 1 + x + 1 ≤ 4, 2x ≤ 4, x ≤ 2
For -1 ≤ x < 1: -x + 1 + x + 1 ≤ 4, 2 ≤ 4 (always true)
For x < -1: -x + 1 - x - 1 ≤ 4, -2x ≤ 4, x
Solve: x² - 4x - 5 < 0
A) -1 < x < 5
B) x < -1 or x > 5
C) -1 < x < 5
D) x < -1 and x > 5
Answer: A) -1 < x < 5
Steps:
Factoring: (x - 5)(x + 1) < 0
The inequality is satisfied when one factor is negative and the other is positive.
x - 5 < 0 and x + 1 > 0, or
x < 5 and x > -1
Therefore, -1 < x < 5
6. Which of the following is equivalent to x > 2 and x < 5?
A) 2 < x < 5
B) x > 2 or x < 5
C) x ≥ 2 and x ≤ 5
D) 2 ≤ x < 5
Answer: A) 2 < x < 5
7. Solve: 2(x - 3) ≥ 4x + 2
A) x ≤ -4
B) x ≥ -4
C) x ≤ 4
D) x ≥ 4
Answer: A) x ≤ -4
Steps:
2x - 6 ≥ 4x + 2
-2x ≥ 8
x ≤ -4
8. What is the solution set of |2x - 1| > 5?
A) x < -2 or x > 3
B) -2 < x < 3
C) x < 3 or x > -2
D) x < -2 and x > 3
Answer: A) x < -2 or x > 3
Steps:
|2x - 1| > 5
2x - 1 < -5 or 2x - 1 > 5
2x < -4 or 2x > 6
x < -2 or x > 3
9. Solve: 3x + 2 < x - 4
A) x > 3
B) x < -3
C) x > -3
D) x < 3
Answer: B) x < -3
Steps:
3x + 2 < x - 4
2x < -6
x < -3
10. Which inequality represents the solution to 5 - 2x ≤ 1?
A) x ≥ 2
B) x ≤ 2
C) x ≥ -2
D) x ≤ -2
Answer: A) x ≥ 2
Steps:
5 - 2x ≤ 1
-2x ≤ -4
x ≥ 2
11. Solve: x²/4 - 1 > 0
A) x < -2 or x > 2
B) -2 < x < 2
C) x ≤ -2 or x ≥ 2
D) -2 ≤ x ≤ 2
Answer: A) x < -2 or x > 2
Steps:
x²/4 > 1
x² > 4
x < -2 or x > 2
12. Which of the following is the solution to |x + 1| + |x - 1| < 4?
A) -1 < x < 1
B) -2 < x < 2
C) -3 < x < 3
D) -4 < x < 4
Answer: B) -2 < x < 2
Steps:
For x ≥ 1: x + 1 + x - 1 < 4, 2x < 4, x < 2
For -1 ≤ x < 1: x + 1 - x + 1 < 4, 2 < 4 (always true)
For x < -1: -x - 1 - x + 1 < 4, -2x < 4, x > -2
Combining these conditions: -2 < x < 2
13. Solve: (x - 2)(x + 3) ≤ 0
A) -3 ≤ x ≤ 2
B) x ≤ -3 or x ≥ 2
C) -3 < x < 2
D) x < -3 or x > 2
Answer: A) -3 ≤ x ≤ 2
Steps:
The product is non-positive when one factor is non-positive and the other is non-negative.
x - 2 ≤ 0 and x + 3 ≥ 0, or
x ≤ 2 and x ≥ -3
Therefore, -3 ≤ x ≤ 2
14. What is the solution set of 2x + 1 < 7 or 3x - 2 ≥ 10?
A) x < 3 or x ≥ 4
B) x > 3 and x < 4
C) x ≤ 3 or x > 4
D) x < 3 and x > 4
Answer: A) x < 3 or x ≥ 4
Steps:
2x + 1 < 7 → x < 3
3x - 2 ≥ 10 → x ≥ 4
Combine: x < 3 or x ≥ 4
15. Solve: √(x + 4) < 3
A) -4 < x < 5
B) -4 ≤ x < 5
C) -4 < x ≤ 5
D) -4 ≤ x ≤ 5
Answer: B) -4 ≤ x < 5
Steps:
√(x + 4) < 3
x + 4 < 9
x < 5
Also, x + 4 ≥ 0 (for real solutions)
x ≥ -4
Combining: -4 ≤ x < 5
16. Which inequality is represented by the graph?
[Image of a number line with an open circle at 2 and shading to the right]
A) x > 2
B) x < 2
C) x ≥ 2
D) x ≤ 2
Answer: A) x > 2
17. Solve: 3(2x - 1) < 4x + 5
A) x > 4
B) x < 4
C) x > -4
D) x < -4
Answer: B) x < 4
Steps:
6x - 3 < 4x + 5
2x < 8
x < 4
18. What is the solution set of |3x - 2| ≥ 4?
A) x ≤ -2/3 or x ≥ 2
B) -2/3 < x < 2
C) x < -2/3 or x > 2
D) x ≤ 2/3 or x ≥ 2
Answer: C) x < -2/3 or x > 2
Steps:
|3x - 2| ≥ 4
3x - 2 ≤ -4 or 3x - 2 ≥ 4
3x ≤ -2 or 3x ≥ 6
x ≤ -2/3 or x ≥ 2
19. Solve: x² + 6x + 8 > 0
A) x < -4 or x > -2
B) -4 < x < -2
C) x ≤ -4 or x ≥ -2
D) -4 ≤ x ≤ -2
Answer: A) x < -4 or x > -2
Steps:
Factoring: (x + 4)(x + 2) > 0
The inequality is satisfied when both factors are positive or both are negative.
x < -4 or x > -2
20. Which of the following is equivalent to 2 < x ≤ 5?
A) x > 2 and x < 5
B) x > 2 and x ≤ 5
C) x ≥ 2 and x ≤ 5
D) x > 2 or x ≤ 5
Answer: B) x > 2 and x ≤ 5
21. Solve: 2x - 3 > 7 or 3x + 1 ≤ 4
A) x > 5 or x ≤ 1
B) x < 5 and x > 1
C) x ≥ 5 or x < 1
D) x > 5 and x < 1
Answer: A) x > 5 or x ≤ 1
Steps:
2x - 3 > 7 → x > 5
3x + 1 ≤ 4 → x ≤ 1
Combine: x > 5 or x ≤ 1
22. What is the solution set of x² - 5x + 6 ≤ 0?
A) 2 ≤ x ≤ 3
B) x < 2 or x > 3
C) 2 < x < 3
D) x ≤ 2 or x ≥ 3
Answer: A) 2 ≤ x ≤ 3
Steps:
Factoring: (x - 2)(x - 3) ≤ 0
The inequality is satisfied when one factor is non-positive and the other is non-negative.
2 ≤ x ≤ 3
23. Solve: 1/x < 1/2
A) x > 2
B) 0 < x < 2
C) x < 0 or x > 2
D) x < 2, x ≠ 0
Answer: C) x < 0 or x > 2
Steps:
1/x < 1/2
2 < x (for x > 0)
2 > x (for x < 0)
Therefore, x < 0 or x > 2
24. Which inequality represents the shaded region?
[Image of a number line with shading to the right of 3, including 3]
A) x > 3
B) x < 3
C) x ≥ 3
D) x ≤ 3
Answer: C) x ≥ 3
25. Solve: |2x + 1| - 3 < 0
A) -2 < x < 1
B) -2 ≤ x ≤ 1
C) -1 < x < 2
D) -1 ≤ x ≤ 2
Answer: A) -2 < x < 1
Steps:
|2x + 1| < 3
-3 < 2x + 1 < 3
-4 < 2x < 2
-2 < x < 1
26. Which of the following is the solution to 3x - 7 ≥ 2x + 5?
A) x ≥ 12
B) x ≤ 12
C) x ≥ -12
D) x ≤ -12
Answer: A) x ≥ 12
Steps:
3x - 7 ≥ 2x + 5
x - 7 ≥ 5
x ≥ 12
27. Solve: x²/9 + y²/4 ≤ 1
A) Ellipse with center (0,0), a = 3, b = 2
B) Ellipse with center (0,0), a = 2, b = 3
C) Circle with center (0,0), radius 1
D) Hyperbola with center (0,0)
Answer: A) Ellipse with center (0,0), a = 3, b = 2
Steps:
This is the standard form of an ellipse equation where a² = 9 and b² = 4.
Therefore, a = 3 and b = 2.
28. What is the solution set of |x - 1| + |x + 1| ≤ 4?
A) -1 ≤ x ≤ 1
B) -2 ≤ x ≤ 2
C) -3 ≤ x ≤ 3
D) -4 ≤ x ≤ 4
Answer: B) -2 ≤ x ≤ 2
Steps:
For x ≥ 1: x - 1 + x + 1 ≤ 4, 2x ≤ 4, x ≤ 2
For -1 ≤ x < 1: -x + 1 + x + 1 ≤ 4, 2 ≤ 4 (always true)
For x < -1: -x + 1 - x - 1 ≤ 4, -2x ≤ 4, x
Solve: x² - 4x - 5 < 0
A) -1 < x < 5
B) x < -1 or x > 5
C) -1 < x < 5
D) x < -1 and x > 5
Answer: A) -1 < x < 5
Steps:
Factoring: (x - 5)(x + 1) < 0
The inequality is satisfied when one factor is negative and the other is positive.
x - 5 < 0 and x + 1 > 0, or
x < 5 and x > -1
Therefore, -1 < x < 5
6. Which of the following is equivalent to x > 2 and x < 5?
A) 2 < x < 5
B) x > 2 or x < 5
C) x ≥ 2 and x ≤ 5
D) 2 ≤ x < 5
Answer: A) 2 < x < 5
7. Solve: 2(x - 3) ≥ 4x + 2
A) x ≤ -4
B) x ≥ -4
C) x ≤ 4
D) x ≥ 4
Answer: A) x ≤ -4
Steps:
2x - 6 ≥ 4x + 2
-2x ≥ 8
x ≤ -4
8. What is the solution set of |2x - 1| > 5?
A) x < -2 or x > 3
B) -2 < x < 3
C) x < 3 or x > -2
D) x < -2 and x > 3
Answer: A) x < -2 or x > 3
Steps:
|2x - 1| > 5
2x - 1 < -5 or 2x - 1 > 5
2x < -4 or 2x > 6
x < -2 or x > 3
9. Solve: 3x + 2 < x - 4
A) x > 3
B) x < -3
C) x > -3
D) x < 3
Answer: B) x < -3
Steps:
3x + 2 < x - 4
2x < -6
x < -3
10. Which inequality represents the solution to 5 - 2x ≤ 1?
A) x ≥ 2
B) x ≤ 2
C) x ≥ -2
D) x ≤ -2
Answer: A) x ≥ 2
Steps:
5 - 2x ≤ 1
-2x ≤ -4
x ≥ 2
11. Solve: x²/4 - 1 > 0
A) x < -2 or x > 2
B) -2 < x < 2
C) x ≤ -2 or x ≥ 2
D) -2 ≤ x ≤ 2
Answer: A) x < -2 or x > 2
Steps:
x²/4 > 1
x² > 4
x < -2 or x > 2
12. Which of the following is the solution to |x + 1| + |x - 1| < 4?
A) -1 < x < 1
B) -2 < x < 2
C) -3 < x < 3
D) -4 < x < 4
Answer: B) -2 < x < 2
Steps:
For x ≥ 1: x + 1 + x - 1 < 4, 2x < 4, x < 2
For -1 ≤ x < 1: x + 1 - x + 1 < 4, 2 < 4 (always true)
For x < -1: -x - 1 - x + 1 < 4, -2x < 4, x > -2
Combining these conditions: -2 < x < 2
13. Solve: (x - 2)(x + 3) ≤ 0
A) -3 ≤ x ≤ 2
B) x ≤ -3 or x ≥ 2
C) -3 < x < 2
D) x < -3 or x > 2
Answer: A) -3 ≤ x ≤ 2
Steps:
The product is non-positive when one factor is non-positive and the other is non-negative.
x - 2 ≤ 0 and x + 3 ≥ 0, or
x ≤ 2 and x ≥ -3
Therefore, -3 ≤ x ≤ 2
14. What is the solution set of 2x + 1 < 7 or 3x - 2 ≥ 10?
A) x < 3 or x ≥ 4
B) x > 3 and x < 4
C) x ≤ 3 or x > 4
D) x < 3 and x > 4
Answer: A) x < 3 or x ≥ 4
Steps:
2x + 1 < 7 → x < 3
3x - 2 ≥ 10 → x ≥ 4
Combine: x < 3 or x ≥ 4
15. Solve: √(x + 4) < 3
A) -4 < x < 5
B) -4 ≤ x < 5
C) -4 < x ≤ 5
D) -4 ≤ x ≤ 5
Answer: B) -4 ≤ x < 5
Steps:
√(x + 4) < 3
x + 4 < 9
x < 5
Also, x + 4 ≥ 0 (for real solutions)
x ≥ -4
Combining: -4 ≤ x < 5
16. Which inequality is represented by the graph?
[Image of a number line with an open circle at 2 and shading to the right]
A) x > 2
B) x < 2
C) x ≥ 2
D) x ≤ 2
Answer: A) x > 2
17. Solve: 3(2x - 1) < 4x + 5
A) x > 4
B) x < 4
C) x > -4
D) x < -4
Answer: B) x < 4
Steps:
6x - 3 < 4x + 5
2x < 8
x < 4
18. What is the solution set of |3x - 2| ≥ 4?
A) x ≤ -2/3 or x ≥ 2
B) -2/3 < x < 2
C) x < -2/3 or x > 2
D) x ≤ 2/3 or x ≥ 2
Answer: C) x < -2/3 or x > 2
Steps:
|3x - 2| ≥ 4
3x - 2 ≤ -4 or 3x - 2 ≥ 4
3x ≤ -2 or 3x ≥ 6
x ≤ -2/3 or x ≥ 2
19. Solve: x² + 6x + 8 > 0
A) x < -4 or x > -2
B) -4 < x < -2
C) x ≤ -4 or x ≥ -2
D) -4 ≤ x ≤ -2
Answer: A) x < -4 or x > -2
Steps:
Factoring: (x + 4)(x + 2) > 0
The inequality is satisfied when both factors are positive or both are negative.
x < -4 or x > -2
20. Which of the following is equivalent to 2 < x ≤ 5?
A) x > 2 and x < 5
B) x > 2 and x ≤ 5
C) x ≥ 2 and x ≤ 5
D) x > 2 or x ≤ 5
Answer: B) x > 2 and x ≤ 5
21. Solve: 2x - 3 > 7 or 3x + 1 ≤ 4
A) x > 5 or x ≤ 1
B) x < 5 and x > 1
C) x ≥ 5 or x < 1
D) x > 5 and x < 1
Answer: A) x > 5 or x ≤ 1
Steps:
2x - 3 > 7 → x > 5
3x + 1 ≤ 4 → x ≤ 1
Combine: x > 5 or x ≤ 1
22. What is the solution set of x² - 5x + 6 ≤ 0?
A) 2 ≤ x ≤ 3
B) x < 2 or x > 3
C) 2 < x < 3
D) x ≤ 2 or x ≥ 3
Answer: A) 2 ≤ x ≤ 3
Steps:
Factoring: (x - 2)(x - 3) ≤ 0
The inequality is satisfied when one factor is non-positive and the other is non-negative.
2 ≤ x ≤ 3
23. Solve: 1/x < 1/2
A) x > 2
B) 0 < x < 2
C) x < 0 or x > 2
D) x < 2, x ≠ 0
Answer: C) x < 0 or x > 2
Steps:
1/x < 1/2
2 < x (for x > 0)
2 > x (for x < 0)
Therefore, x < 0 or x > 2
24. Which inequality represents the shaded region?
[Image of a number line with shading to the right of 3, including 3]
A) x > 3
B) x < 3
C) x ≥ 3
D) x ≤ 3
Answer: C) x ≥ 3
25. Solve: |2x + 1| - 3 < 0
A) -2 < x < 1
B) -2 ≤ x ≤ 1
C) -1 < x < 2
D) -1 ≤ x ≤ 2
Answer: A) -2 < x < 1
Steps:
|2x + 1| < 3
-3 < 2x + 1 < 3
-4 < 2x < 2
-2 < x < 1
26. Which of the following is the solution to 3x - 7 ≥ 2x + 5?
A) x ≥ 12
B) x ≤ 12
C) x ≥ -12
D) x ≤ -12
Answer: A) x ≥ 12
Steps:
3x - 7 ≥ 2x + 5
x - 7 ≥ 5
x ≥ 12
27. Solve: x²/9 + y²/4 ≤ 1
A) Ellipse with center (0,0), a = 3, b = 2
B) Ellipse with center (0,0), a = 2, b = 3
C) Circle with center (0,0), radius 1
D) Hyperbola with center (0,0)
Answer: A) Ellipse with center (0,0), a = 3, b = 2
Steps:
This is the standard form of an ellipse equation where a² = 9 and b² = 4.
Therefore, a = 3 and b = 2.
28. What is the solution set of |x - 1| + |x + 1| ≤ 4?
A) -1 ≤ x ≤ 1
B) -2 ≤ x ≤ 2
C) -3 ≤ x ≤ 3
D) -4 ≤ x ≤ 4
Answer: B) -2 ≤ x ≤ 2
Steps:
For x ≥ 1: x - 1 + x + 1 ≤ 4, 2x ≤ 4, x ≤ 2
For -1 ≤ x < 1: -x + 1 + x + 1 ≤ 4, 2 ≤ 4 (always true)
For x < -1: -x + 1 - x - 1 ≤ 4, -2x ≤ 4, x
Solve: x² - 4x - 5 < 0
A) -1 < x < 5
B) x < -1 or x > 5
C) -1 < x < 5
D) x < -1 and x > 5
Answer: A) -1 < x < 5
Steps:
Factoring: (x - 5)(x + 1) < 0
The inequality is satisfied when one factor is negative and the other is positive.
x - 5 < 0 and x + 1 > 0, or
x < 5 and x > -1
Therefore, -1 < x < 5
6. Which of the following is equivalent to x > 2 and x < 5?
A) 2 < x < 5
B) x > 2 or x < 5
C) x ≥ 2 and x ≤ 5
D) 2 ≤ x < 5
Answer: A) 2 < x < 5
7. Solve: 2(x - 3) ≥ 4x + 2
A) x ≤ -4
B) x ≥ -4
C) x ≤ 4
D) x ≥ 4
Answer: A) x ≤ -4
Steps:
2x - 6 ≥ 4x + 2
-2x ≥ 8
x ≤ -4
8. What is the solution set of |2x - 1| > 5?
A) x < -2 or x > 3
B) -2 < x < 3
C) x < 3 or x > -2
D) x < -2 and x > 3
Answer: A) x < -2 or x > 3
Steps:
|2x - 1| > 5
2x - 1 < -5 or 2x - 1 > 5
2x < -4 or 2x > 6
x < -2 or x > 3
9. Solve: 3x + 2 < x - 4
A) x > 3
B) x < -3
C) x > -3
D) x < 3
Answer: B) x < -3
Steps:
3x + 2 < x - 4
2x < -6
x < -3
10. Which inequality represents the solution to 5 - 2x ≤ 1?
A) x ≥ 2
B) x ≤ 2
C) x ≥ -2
D) x ≤ -2
Answer: A) x ≥ 2
Steps:
5 - 2x ≤ 1
-2x ≤ -4
x ≥ 2
11. Solve: x²/4 - 1 > 0
A) x < -2 or x > 2
B) -2 < x < 2
C) x ≤ -2 or x ≥ 2
D) -2 ≤ x ≤ 2
Answer: A) x < -2 or x > 2
Steps:
x²/4 > 1
x² > 4
x < -2 or x > 2
12. Which of the following is the solution to |x + 1| + |x - 1| < 4?
A) -1 < x < 1
B) -2 < x < 2
C) -3 < x < 3
D) -4 < x < 4
Answer: B) -2 < x < 2
Steps:
For x ≥ 1: x + 1 + x - 1 < 4, 2x < 4, x < 2
For -1 ≤ x < 1: x + 1 - x + 1 < 4, 2 < 4 (always true)
For x < -1: -x - 1 - x + 1 < 4, -2x < 4, x > -2
Combining these conditions: -2 < x < 2
13. Solve: (x - 2)(x + 3) ≤ 0
A) -3 ≤ x ≤ 2
B) x ≤ -3 or x ≥ 2
C) -3 < x < 2
D) x < -3 or x > 2
Answer: A) -3 ≤ x ≤ 2
Steps:
The product is non-positive when one factor is non-positive and the other is non-negative.
x - 2 ≤ 0 and x + 3 ≥ 0, or
x ≤ 2 and x ≥ -3
Therefore, -3 ≤ x ≤ 2
14. What is the solution set of 2x + 1 < 7 or 3x - 2 ≥ 10?
A) x < 3 or x ≥ 4
B) x > 3 and x < 4
C) x ≤ 3 or x > 4
D) x < 3 and x > 4
Answer: A) x < 3 or x ≥ 4
Steps:
2x + 1 < 7 → x < 3
3x - 2 ≥ 10 → x ≥ 4
Combine: x < 3 or x ≥ 4
15. Solve: √(x + 4) < 3
A) -4 < x < 5
B) -4 ≤ x < 5
C) -4 < x ≤ 5
D) -4 ≤ x ≤ 5
Answer: B) -4 ≤ x < 5
Steps:
√(x + 4) < 3
x + 4 < 9
x < 5
Also, x + 4 ≥ 0 (for real solutions)
x ≥ -4
Combining: -4 ≤ x < 5
16. Which inequality is represented by the graph?
[Image of a number line with an open circle at 2 and shading to the right]
A) x > 2
B) x < 2
C) x ≥ 2
D) x ≤ 2
Answer: A) x > 2
17. Solve: 3(2x - 1) < 4x + 5
A) x > 4
B) x < 4
C) x > -4
D) x < -4
Answer: B) x < 4
Steps:
6x - 3 < 4x + 5
2x < 8
x < 4
18. What is the solution set of |3x - 2| ≥ 4?
A) x ≤ -2/3 or x ≥ 2
B) -2/3 < x < 2
C) x < -2/3 or x > 2
D) x ≤ 2/3 or x ≥ 2
Answer: C) x < -2/3 or x > 2
Steps:
|3x - 2| ≥ 4
3x - 2 ≤ -4 or 3x - 2 ≥ 4
3x ≤ -2 or 3x ≥ 6
x ≤ -2/3 or x ≥ 2
19. Solve: x² + 6x + 8 > 0
A) x < -4 or x > -2
B) -4 < x < -2
C) x ≤ -4 or x ≥ -2
D) -4 ≤ x ≤ -2
Answer: A) x < -4 or x > -2
Steps:
Factoring: (x + 4)(x + 2) > 0
The inequality is satisfied when both factors are positive or both are negative.
x < -4 or x > -2
20. Which of the following is equivalent to 2 < x ≤ 5?
A) x > 2 and x < 5
B) x > 2 and x ≤ 5
C) x ≥ 2 and x ≤ 5
D) x > 2 or x ≤ 5
Answer: B) x > 2 and x ≤ 5
21. Solve: 2x - 3 > 7 or 3x + 1 ≤ 4
A) x > 5 or x ≤ 1
B) x < 5 and x > 1
C) x ≥ 5 or x < 1
D) x > 5 and x < 1
Answer: A) x > 5 or x ≤ 1
Steps:
2x - 3 > 7 → x > 5
3x + 1 ≤ 4 → x ≤ 1
Combine: x > 5 or x ≤ 1
22. What is the solution set of x² - 5x + 6 ≤ 0?
A) 2 ≤ x ≤ 3
B) x < 2 or x > 3
C) 2 < x < 3
D) x ≤ 2 or x ≥ 3
Answer: A) 2 ≤ x ≤ 3
Steps:
Factoring: (x - 2)(x - 3) ≤ 0
The inequality is satisfied when one factor is non-positive and the other is non-negative.
2 ≤ x ≤ 3
23. Solve: 1/x < 1/2
A) x > 2
B) 0 < x < 2
C) x < 0 or x > 2
D) x < 2, x ≠ 0
Answer: C) x < 0 or x > 2
Steps:
1/x < 1/2
2 < x (for x > 0)
2 > x (for x < 0)
Therefore, x < 0 or x > 2
24. Which inequality represents the shaded region?
[Image of a number line with shading to the right of 3, including 3]
A) x > 3
B) x < 3
C) x ≥ 3
D) x ≤ 3
Answer: C) x ≥ 3
25. Solve: |2x + 1| - 3 < 0
A) -2 < x < 1
B) -2 ≤ x ≤ 1
C) -1 < x < 2
D) -1 ≤ x ≤ 2
Answer: A) -2 < x < 1
Steps:
|2x + 1| < 3
-3 < 2x + 1 < 3
-4 < 2x < 2
-2 < x < 1
26. Which of the following is the solution to 3x - 7 ≥ 2x + 5?
A) x ≥ 12
B) x ≤ 12
C) x ≥ -12
D) x ≤ -12
Answer: A) x ≥ 12
Steps:
3x - 7 ≥ 2x + 5
x - 7 ≥ 5
x ≥ 12
27. Solve: x²/9 + y²/4 ≤ 1
A) Ellipse with center (0,0), a = 3, b = 2
B) Ellipse with center (0,0), a = 2, b = 3
C) Circle with center (0,0), radius 1
D) Hyperbola with center (0,0)
Answer: A) Ellipse with center (0,0), a = 3, b = 2
Steps:
This is the standard form of an ellipse equation where a² = 9 and b² = 4.
Therefore, a = 3 and b = 2.
28. What is the solution set of |x - 1| + |x + 1| ≤ 4?
A) -1 ≤ x ≤ 1
B) -2 ≤ x ≤ 2
C) -3 ≤ x ≤ 3
D) -4 ≤ x ≤ 4
Answer: B) -2 ≤ x ≤ 2
Steps:
For x ≥ 1: x - 1 + x + 1 ≤ 4, 2x ≤ 4, x ≤ 2
For -1 ≤ x < 1: -x + 1 + x + 1 ≤ 4, 2 ≤ 4 (always true)
For x < -1: -x + 1 - x - 1 ≤ 4, -2x ≤ 4, x
