Algebra (Secondary) for Competitive Exam Success

Algebra (Secondary)

3D Geometry Arithmetic and Geometric Progressions Arithmetic and Geometric Progressions - Applications Arithmetic and Geometric Progressions - Case Studies Arithmetic and Geometric Progressions - Problem Set Binomial Theorem Binomial Theorem (Intro) Binomial Theorem (Intro) - Applications Binomial Theorem (Intro) - Case Studies Binomial Theorem (Intro) - Problem Set Complex Numbers and Quadratic Equations Coordinate Geometry - Circles Coordinate Geometry - Straight Lines Differential Equations Factorization Techniques Factorization Techniques - Applications Factorization Techniques - Case Studies Factorization Techniques - Problem Set Indices and Surds Indices and Surds - Applications Indices and Surds - Case Studies Indices and Surds - Problem Set Inequalities and Their Applications Inequalities and Their Applications - Applications Inequalities and Their Applications - Case Studies Inequalities and Their Applications - Problem Set Integral Calculus Limit, Continuity and Differentiability Linear Equations in One Variable Linear Equations in One Variable - Applications Linear Equations in One Variable - Case Studies Linear Equations in One Variable - Problem Set Linear Programming Basics Linear Programming Basics - Applications Linear Programming Basics - Case Studies Linear Programming Basics - Problem Set Mathematical Induction Matrices and Determinants Pair of Linear Equations - Word Problems Pair of Linear Equations - Word Problems - Applications Pair of Linear Equations - Word Problems - Case Studies Pair of Linear Equations - Word Problems - Problem Set Permutations and Combinations Polynomials - Introduction Polynomials - Introduction - Applications Polynomials - Introduction - Case Studies Polynomials - Introduction - Problem Set Polynomials - Roots and Factor Theorem Polynomials - Roots and Factor Theorem - Applications Polynomials - Roots and Factor Theorem - Case Studies Polynomials - Roots and Factor Theorem - Problem Set Quadratic Equations Quadratic Equations - Applications Quadratic Equations - Case Studies Quadratic Equations - Problem Set Quadratic Formula Applications Quadratic Formula Applications - Applications Quadratic Formula Applications - Case Studies Quadratic Formula Applications - Problem Set Sequences and Series Sets, Relations and Functions Simultaneous Linear Equations Simultaneous Linear Equations - Applications Simultaneous Linear Equations - Case Studies Simultaneous Linear Equations - Problem Set Trigonometry - Advanced Problems Vector Algebra

Q. Which of the following is a solution to the inequality: -4x + 1 ≤ 9?

A. x ≤ -2
B. x ≥ -2
C. x ≤ 2
D. x ≥ 2

Solution

Step 1: Subtract 1 from both sides: -4x ≤ 8. Step 2: Divide by -4 (reverse inequality): x ≥ -2.

Correct Answer: A — x ≤ -2

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Q. Which of the following is a solution to the inequality: 2(x - 3) > 4?

A. x = 5
B. x = 4
C. x = 3
D. x = 6

Solution

Step 1: Divide by 2: x - 3 > 2. Step 2: Add 3 to both sides: x > 5. Thus, x = 5 is a solution.

Correct Answer: A — x = 5

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Q. Which of the following is a solution to the inequality: 2x + 3 > 11?

A. x < 4
B. x > 4
C. x = 4
D. x = 5

Solution

Step 1: Subtract 3 from both sides: 2x > 8. Step 2: Divide by 2: x > 4.

Correct Answer: B — x > 4

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Q. Which of the following is a solution to the inequality: 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 by 2: x ≥ 4.

Correct Answer: C — x = 4

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Q. Which of the following is a solution to the inequality: 3x + 2 < 11?

A. x = 1
B. x = 2
C. x = 3
D. x = 4

Solution

Step 1: Subtract 2 from both sides: 3x < 9. Step 2: Divide by 3: x < 3.

Correct Answer: B — x = 2

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Q. Which of the following is a solution to the inequality: 4x - 1 ≤ 3x + 2?

A. x ≤ 3
B. x ≥ 3
C. x < 3
D. x > 3

Solution

Step 1: Subtract 3x from both sides: x - 1 ≤ 2. Step 2: Add 1: x ≤ 3.

Correct Answer: A — x ≤ 3

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Q. Which of the following is a solution to the inequality: 6 - 2x ≤ 4?

A. x ≥ 1
B. x ≤ 1
C. x ≥ 2
D. x ≤ 2

Solution

Step 1: Subtract 6 from both sides: -2x ≤ -2. Step 2: Divide by -2 (reverse the inequality): x ≥ 1.

Correct Answer: B — x ≤ 1

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Q. Which of the following is a solution to the inequality: 7 - 2x > 1?

A. x < 3
B. x > 3
C. x ≤ 3
D. x ≥ 3

Solution

Step 1: Subtract 7 from both sides: -2x > -6. Step 2: Divide by -2 (reverse the inequality): x < 3.

Correct Answer: A — x < 3

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Q. Which of the following is a solution to the inequality: 7 - 3x < 1?

A. x > 2
B. x < 2
C. x > 1
D. x < 1

Solution

Step 1: Subtract 7 from both sides: -3x < -6. Step 2: Divide by -3 (reverse the inequality): x > 2.

Correct Answer: A — x > 2

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Q. Which of the following is a solution to the inequality: 7 - x > 2?

A. x < 5
B. x > 5
C. x < 7
D. x > 7

Solution

Step 1: Subtract 7 from both sides: -x > -5. Step 2: Multiply by -1 (reverse the inequality): x < 5.

Correct Answer: A — x < 5

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Q. Which of the following is a solution to the inequality: x^2 + 2x - 8 < 0?

A. x < -4
B. -4 < x < 2
C. x > 2
D. x = -4

Solution

Step 1: Factor: (x + 4)(x - 2) < 0. Step 2: Critical points are x = -4 and x = 2. Step 3: Test intervals: valid for -4 < x < 2.

Correct Answer: B — -4 < x < 2

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Q. Which of the following is a solution to the inequality: x^2 - 4x < 0?

A. x < 0
B. 0 < x < 4
C. x > 4
D. x = 2

Solution

Step 1: Factor the inequality: x(x - 4) < 0. Step 2: The critical points are x = 0 and x = 4. Step 3: Test intervals: (0, 4) is valid.

Correct Answer: B — 0 < x < 4

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Q. Which of the following is the correct factored form of the quadratic x^2 - 5x + 6?

A. (x - 2)(x - 3)
B. (x + 2)(x + 3)
C. (x - 1)(x - 6)
D. (x + 1)(x + 6)

Solution

To factor, find two numbers that multiply to 6 and add to -5.\nThe numbers are -2 and -3.\nThus, the factored form is (x - 2)(x - 3).

Correct Answer: A — (x - 2)(x - 3)

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Q. Which of the following is the correct factorization of 2x^2 + 8x?

A. 2x(x + 4)
B. 2(x^2 + 4x)
C. x(2x + 8)
D. 2x^2(1 + 4)

Solution

First, factor out the greatest common factor, which is 2x. This gives us 2x(x + 4).

Correct Answer: A — 2x(x + 4)

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Q. Which of the following is the correct factorization of x^2 + 5x + 6?

A. (x + 2)(x + 3)
B. (x - 2)(x - 3)
C. (x + 1)(x + 6)
D. (x - 1)(x - 6)

Solution

Step 1: Find two numbers that multiply to 6 and add to 5: 2 and 3. Step 2: Factor as (x + 2)(x + 3).

Correct Answer: A — (x + 2)(x + 3)

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Q. Which of the following is the correct factorization of 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

The polynomial factors to (x + 5)(x + 2) since 5 and 2 add to 7 and multiply to 10.

Correct Answer: A — (x + 5)(x + 2)

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Q. Which of the following is the correct factorization of x^2 - 6x + 9?

A. (x - 3)(x - 3)
B. (x + 3)(x + 3)
C. (x - 9)(x + 1)
D. (x + 6)(x - 3)

Solution

The polynomial can be factored as (x - 3)(x - 3) or (x - 3)^2.

Correct Answer: A — (x - 3)(x - 3)

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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).

Correct Answer: A — (x - 3)(x + 3)

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Q. Which of the following is the factored form of 2x^2 + 8x?

A. 2x(x + 4)
B. 2(x + 4)(x + 2)
C. x(2x + 8)
D. 2x(x + 2)

Solution

First, factor out the greatest common factor, which is 2x. This gives us 2x(x + 4).

Correct Answer: A — 2x(x + 4)

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Q. Which of the following is the factored form of the quadratic equation x^2 - 5x + 6?

A. (x - 2)(x - 3)
B. (x + 2)(x + 3)
C. (x - 1)(x - 6)
D. (x + 1)(x + 6)

Solution

To factor x^2 - 5x + 6, find two numbers that multiply to 6 and add to -5.\nThe numbers are -2 and -3.\nThus, the factored form is (x - 2)(x - 3).

Correct Answer: A — (x - 2)(x - 3)

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Q. Which of the following is the factored form of x^2 - 5x + 6?

A. (x - 2)(x - 3)
B. (x + 2)(x + 3)
C. (x - 1)(x - 6)
D. (x + 1)(x + 6)

Solution

To factor x^2 - 5x + 6, find two numbers that multiply to 6 and add to -5.\nThe numbers are -2 and -3.\nThus, the factored form is (x - 2)(x - 3).

Correct Answer: A — (x - 2)(x - 3)

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Q. Which of the following is the result of (x + 2)(x - 2)?

A. x^2 - 4
B. x^2 + 4
C. x^2 - 2
D. x^2 + 2

Solution

Use the difference of squares: (x + 2)(x - 2) = x^2 - 4.

Correct Answer: A — x^2 - 4

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Q. Which of the following is the result of factoring x^2 + 5x + 6?

A. (x + 2)(x + 3)
B. (x - 2)(x - 3)
C. (x + 1)(x + 6)
D. (x - 1)(x - 6)

Solution

The polynomial factors to (x + 2)(x + 3) since 2 and 3 multiply to 6 and add to 5.

Correct Answer: A — (x + 2)(x + 3)

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Q. Which of the following is the solution set for the inequality 2x + 1 ≥ 5?

A. x ≤ 2
B. x ≥ 2
C. x < 2
D. x > 2

Solution

Step 1: Subtract 1 from both sides: 2x ≥ 4. Step 2: Divide by 2: x ≥ 2.

Correct Answer: B — x ≥ 2

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Q. Which of the following is the solution set for the inequality 2x + 3 > 7?

A. x < 2
B. x > 2
C. x < 3
D. x > 3

Solution

Step 1: Subtract 3 from both sides: 2x > 4. Step 2: Divide by 2: x > 2.

Correct Answer: B — x > 2

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Q. Which of the following is the solution set for the inequality x + 2 > 3?

A. x < 1
B. x > 1
C. x < 5
D. x > 5

Solution

To solve the inequality, subtract 2 from both sides: x > 1.

Correct Answer: B — x > 1

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Q. Which of the following is the solution set for the inequality x + 3 > 2?

A. x > -1
B. x < -1
C. x > 1
D. x < 1

Solution

To solve the inequality, subtract 3 from both sides: x > -1.

Correct Answer: A — x > -1

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Q. Which of the following is the solution set for the inequality x + 4 > 2?

A. x > -2
B. x < -2
C. x > 6
D. x < 6

Solution

Step 1: Subtract 4 from both sides: x > -2.

Correct Answer: A — x > -2

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Q. Which of the following is the solution set for the inequality x^2 - 4 > 0?

A. (-∞, -2) ∪ (2, ∞)
B. (-2, 2)
C. [-2, 2]
D. [2, ∞)

Solution

Factor: (x - 2)(x + 2) > 0. The solution is x < -2 or x > 2.

Correct Answer: A — (-∞, -2) ∪ (2, ∞)

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Q. Which of the following is the solution to the equation 3x + 4 = 10?

A. x = 2
B. x = 3
C. x = 4
D. x = 1

Solution

Subtract 4 from both sides: 3x = 6. Then divide by 3: x = 2.

Correct Answer: A — x = 2

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Algebra (Secondary)

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