Step 1: Subtract 3 from both sides: 2x ≥ 4. Step 2: Divide both sides by 2: x ≥ 2.
Correct Answer: A — x ≥ 2
Q. Solve the inequality 3x - 5 < 4.
A.x < 3
B.x < 2
C.x > 3
D.x > 2
Solution
Step 1: Add 5 to both sides: 3x < 9. Step 2: Divide by 3: x < 3.
Correct Answer: B — x < 2
Q. Solve the inequality 5x - 2 < 3.
A.x < 1
B.x > 1
C.x < 2
D.x > 2
Solution
Step 1: Add 2 to both sides: 5x < 5. Step 2: Divide both sides by 5: x < 1.
Correct Answer: A — x < 1
Q. What are the roots of the quadratic equation x^2 - 5x + 6 = 0?
A.x = 1, 6
B.x = 2, 3
C.x = -2, -3
D.x = 0, 6
Solution
Step 1: Factor the equation: (x - 2)(x - 3) = 0. Step 2: Set each factor to zero: x - 2 = 0 or x - 3 = 0. Step 3: Solutions are x = 2 and x = 3.
Correct Answer: B — x = 2, 3
Q. What are the solutions to the quadratic equation x^2 + 4x + 4 = 0?
A.x = -2, -2
B.x = 2, 2
C.x = -4, 0
D.x = 0, 4
Solution
Step 1: Factor the equation: (x + 2)(x + 2) = 0. Step 2: Set the factor to zero: x + 2 = 0. Step 3: Solution is x = -2 (double root).
Correct Answer: A — x = -2, -2
Q. What is the factored form of the polynomial x^2 + 5x + 6?
A.(x + 2)(x + 3)
B.(x - 2)(x - 3)
C.(x + 1)(x + 6)
D.(x + 3)(x + 2)
Solution
Step 1: Find two numbers that multiply to 6 and add to 5: 2 and 3. Step 2: Factor: (x + 2)(x + 3).
Correct Answer: A — (x + 2)(x + 3)
Q. What is the factored form of the polynomial x^2 - 9?
A.(x - 3)(x + 3)
B.(x - 9)(x + 1)
C.(x - 1)(x + 1)
D.(x + 3)(x + 3)
Solution
Step 1: Recognize it as a difference of squares: x^2 - 3^2. Step 2: Factor: (x - 3)(x + 3).
Correct Answer: A — (x - 3)(x + 3)
Q. What is the result of factoring the polynomial x^2 + 7x + 10?
A.(x + 5)(x + 2)
B.(x + 10)(x - 1)
C.(x - 5)(x - 2)
D.(x + 1)(x + 10)
Solution
Step 1: Find two numbers that multiply to 10 and add to 7: 5 and 2. Step 2: Factor: (x + 5)(x + 2).
Correct Answer: A — (x + 5)(x + 2)
Q. What is the solution set for the inequality 2x + 5 > 3?
A.x > -1
B.x < -1
C.x > 1
D.x < 1
Solution
Step 1: Subtract 5 from both sides: 2x > -2. Step 2: Divide by 2: x > -1.
Correct Answer: A — x > -1
Q. What is the solution to the equation 2x + 3 = 11?
A.x = 2
B.x = 3
C.x = 4
D.x = 5
Solution
Step 1: Subtract 3 from both sides: 2x = 8. Step 2: Divide both sides by 2: x = 4.
Correct Answer: C — x = 4
Q. What is the solution to the equation 3(x - 1) = 12?
A.x = 3
B.x = 4
C.x = 5
D.x = 6
Solution
Step 1: Divide both sides by 3: x - 1 = 4. Step 2: Add 1 to both sides: x = 5.
Correct Answer: B — x = 4
Q. What is the solution to the equation 4x - 12 = 0?
A.x = 3
B.x = 4
C.x = 2
D.x = 0
Solution
Step 1: Add 12 to both sides: 4x = 12. Step 2: Divide by 4: x = 3.
Correct Answer: A — x = 3
Q. What is the solution to the equation 4x - 7 = 5?
A.x = 1
B.x = 2
C.x = 3
D.x = 4
Solution
Step 1: Add 7 to both sides: 4x = 12. Step 2: Divide both sides by 4: x = 3.
Correct Answer: B — x = 2
Q. What is the solution to the equation 5x + 2 = 17?
A.x = 3
B.x = 4
C.x = 5
D.x = 2
Solution
Step 1: Subtract 2 from both sides: 5x = 15. Step 2: Divide by 5: x = 3.
Correct Answer: B — x = 4
Q. What is the solution to the equation 5x - 2 = 3x + 6?
A.x = 2
B.x = 3
C.x = 4
D.x = 5
Solution
Step 1: Subtract 3x from both sides: 2x - 2 = 6. Step 2: Add 2 to both sides: 2x = 8. Step 3: Divide by 2: x = 4.
Correct Answer: B — x = 3
Q. What is the solution to the equation 7x + 2 = 3x + 18?
A.x = 4
B.x = 3
C.x = 5
D.x = 2
Solution
Step 1: Subtract 3x from both sides: 4x + 2 = 18. Step 2: Subtract 2: 4x = 16. Step 3: Divide by 4: x = 4.
Correct Answer: A — x = 4
Q. What is the solution to the quadratic equation x^2 + 4x + 4 = 0?
A.x = -2
B.x = 2
C.x = 0
D.x = -4
Solution
Step 1: Factor the equation: (x + 2)(x + 2) = 0. Step 2: Set the factor to zero: x + 2 = 0. Solution is x = -2.
Correct Answer: A — x = -2
Q. What is the sum of the roots of the quadratic equation x^2 - 6x + 8 = 0?
A.6
B.8
C.4
D.2
Solution
Step 1: Use the sum of roots formula -b/a. Here, b = -6, a = 1. Step 2: Sum = 6.
Correct Answer: A — 6
Q. What is the value of x in the equation 4(x - 1) = 2(x + 3)?
A.x = 5
B.x = 4
C.x = 3
D.x = 2
Solution
Step 1: Distribute: 4x - 4 = 2x + 6. Step 2: Subtract 2x from both sides: 2x - 4 = 6. Step 3: Add 4: 2x = 10. Step 4: Divide by 2: x = 5.
Correct Answer: C — x = 3
Q. What is the value of x in the inequality 3x - 7 < 2?
A.x < 3
B.x < 2
C.x > 3
D.x > 2
Solution
Step 1: Add 7 to both sides: 3x < 9. Step 2: Divide by 3: x < 3.
Correct Answer: A — x < 3
Q. What is the value of x in the polynomial equation 3x^2 + 12x = 0?
A.x = 0, -4
B.x = 0, 4
C.x = -3, -4
D.x = 3, 4
Solution
Step 1: Factor out 3x: 3x(x + 4) = 0. Step 2: Set each factor to zero: 3x = 0 or x + 4 = 0. Step 3: Solutions are x = 0 and x = -4.
Correct Answer: A — x = 0, -4
Q. What is the vertex of the parabola represented by the equation y = x^2 - 4x + 3?
A.(2, -1)
B.(2, 1)
C.(1, 2)
D.(3, 0)
Solution
Step 1: Use the vertex formula x = -b/2a: x = 4/2 = 2. Step 2: Substitute x back into the equation: y = 2^2 - 4(2) + 3 = -1. Vertex is (2, -1).
Correct Answer: A — (2, -1)
Q. What is the vertex of the quadratic function y = x^2 - 4x + 3?
A.(2, -1)
B.(2, -4)
C.(1, 2)
D.(3, 0)
Solution
Step 1: Use the vertex formula x = -b/2a. Here, a = 1, b = -4. Step 2: x = 4/2 = 2. Step 3: Substitute x back: y = 2^2 - 4*2 + 3 = -1. Vertex is (2, -1).
Correct Answer: A — (2, -1)
Q. Which of the following is a factor of the polynomial x^2 - 5x + 6?
A.(x - 2)
B.(x - 3)
C.(x + 2)
D.(x + 3)
Solution
Step 1: Factor the polynomial: (x - 2)(x - 3). Step 2: The factors are (x - 2) and (x - 3).
Correct Answer: A — (x - 2)
Q. Which of the following is the correct factorization of x^2 - 9?
A.(x - 3)(x + 3)
B.(x - 3)(x - 3)
C.(x + 3)(x + 3)
D.(x - 9)(x + 1)
Solution
Step 1: Recognize it as a difference of squares: a^2 - b^2 = (a - b)(a + b). Step 2: Here, a = x, b = 3. So, (x - 3)(x + 3).
