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. Solve the inequality: -3x + 4 > 2.

A. x < 2/3
B. x > 2/3
C. x ≤ 2/3
D. x ≥ 2/3

Solution

Step 1: Subtract 4 from both sides: -3x > -2. Step 2: Divide by -3 (reverse the inequality): x < 2/3.

Correct Answer: B — x > 2/3

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Q. Solve the inequality: -3x + 7 ≤ 1.

A. x ≥ 2
B. x ≤ 2
C. x > 2
D. x < 2

Solution

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

Correct Answer: B — x ≤ 2

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Q. Solve 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. Solve the inequality: 2(x - 1) ≥ 3x + 4.

A. x ≤ -2
B. x ≥ -2
C. x ≤ 2
D. x ≥ 2

Solution

Step 1: Distribute: 2x - 2 ≥ 3x + 4. Step 2: Subtract 2x from both sides: -2 ≥ x + 4. Step 3: Subtract 4: -6 ≥ x. Step 4: Reverse: x ≤ -6.

Correct Answer: A — x ≤ -2

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Q. Solve the inequality: 2(x - 3) ≤ 4.

A. x ≤ 5
B. x ≥ 5
C. x < 5
D. x > 5

Solution

Step 1: Distribute: 2x - 6 ≤ 4. Step 2: Add 6 to both sides: 2x ≤ 10. Step 3: Divide by 2: x ≤ 5.

Correct Answer: A — x ≤ 5

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Q. Solve the inequality: 2x - 3 < 7

A. x < 5
B. x > 5
C. x < 2
D. x > 2

Solution

Add 3 to both sides: 2x < 10. Then divide by 2: x < 5.

Correct Answer: A — x < 5

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Q. Solve the inequality: 2x - 5 < 3

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

Solution

Add 5 to both sides: 2x < 8. Divide by 2: x < 4.

Correct Answer: A — x < 4

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Q. Solve the inequality: 3x + 4 > 10

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

Solution

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

Correct Answer: A — x > 2

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Q. Solve the inequality: 4x + 1 ≥ 2x + 9.

A. x ≥ 4
B. x ≤ 4
C. x ≥ 2
D. x ≤ 2

Solution

Step 1: Subtract 2x from both sides: 2x + 1 ≥ 9. Step 2: Subtract 1: 2x ≥ 8. Step 3: Divide by 2: x ≥ 4.

Correct Answer: A — x ≥ 4

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Q. Solve the inequality: 4x - 7 > 2x + 5.

A. x > 6
B. x < 6
C. x > 5
D. x < 5

Solution

Step 1: Subtract 2x from both sides: 2x - 7 > 5. Step 2: Add 7: 2x > 12. Step 3: Divide by 2: x > 6.

Correct Answer: A — x > 6

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Q. Solve the inequality: 6 - 3x < 0.

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

Solution

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

Correct Answer: A — x > 2

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Q. Solve the inequality: 7 - 2x < 1.

A. x > 3
B. x < 3
C. x > 4
D. x < 4

Solution

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

Correct Answer: B — x < 3

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Q. Solve the inequality: 7 - 2x ≥ 3.

A. x ≤ 2
B. x ≥ 2
C. x ≤ 3
D. x ≥ 3

Solution

Step 1: Subtract 7 from both sides: -2x ≥ -4. Step 2: Divide by -2 (reverse inequality): x ≤ 2.

Correct Answer: B — x ≥ 2

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Q. Solve the inequality: 7 - 3x < 1.

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

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. Solve the inequality: 7 - 3x ≥ 1.

A. x ≤ 2
B. x ≥ 2
C. x ≤ 3
D. x ≥ 3

Solution

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

Correct Answer: B — x ≥ 2

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Q. Solve the inequality: 7x + 1 ≤ 3x + 9.

A. x ≤ 2
B. x ≥ 2
C. x ≤ 4
D. x ≥ 4

Solution

Step 1: Subtract 3x from both sides: 4x + 1 ≤ 9. Step 2: Subtract 1: 4x ≤ 8. Step 3: Divide by 4: x ≤ 2.

Correct Answer: A — x ≤ 2

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Q. Solve the inequality: 7x + 2 ≤ 3x + 10.

A. x ≤ 2
B. x ≥ 2
C. x < 2
D. x > 2

Solution

Step 1: Subtract 3x from both sides: 4x + 2 ≤ 10. Step 2: Subtract 2: 4x ≤ 8. Step 3: Divide by 4: x ≥ 2.

Correct Answer: B — x ≥ 2

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Q. Solve the inequality: x^2 + 2x - 8 < 0.

A. -4 < x < 2
B. x < -4 or x > 2
C. x > -4 and x < 2
D. x < 2

Solution

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

Correct Answer: A — -4 < x < 2

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Q. Solve the inequality: x^2 + 4x < 12.

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

Solution

Step 1: Rearrange: x^2 + 4x - 12 < 0. Step 2: Factor: (x + 6)(x - 2) < 0. Step 3: The solution is -6 < x < 2.

Correct Answer: D — x > -6 and x < 2

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Q. Solve the inequality: x^2 - 4 > 0.

A. x < -2 or x > 2
B. x > -2 and x < 2
C. x < 2
D. x > 2

Solution

Step 1: Factor: (x - 2)(x + 2) > 0. Step 2: Test intervals: solution is x < -2 or x > 2.

Correct Answer: A — x < -2 or x > 2

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Q. Solve the inequality: x^2 - 4x < 0.

A. x < 0 or x > 4
B. 0 < x < 4
C. x > 0
D. x < 4

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 the solution.

Correct Answer: B — 0 < x < 4

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Q. Solve the inequality: x^2 - 5x + 6 > 0.

A. (2, 3)
B. (3, ∞)
C. (-∞, 2) ∪ (3, ∞)
D. (2, ∞)

Solution

Step 1: Factor: (x - 2)(x - 3) > 0. Step 2: Test intervals: solution is (-∞, 2) ∪ (3, ∞).

Correct Answer: C — (-∞, 2) ∪ (3, ∞)

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Q. Solve the inequality: x^2 - 5x + 6 ≤ 0.

A. x ≤ 2 or x ≥ 3
B. x ≥ 2 and x ≤ 3
C. x < 2 or x > 3
D. x > 2 and x < 3

Solution

Step 1: Factor the quadratic: (x - 2)(x - 3) ≤ 0. Step 2: The solution is 2 ≤ x ≤ 3.

Correct Answer: B — x ≥ 2 and x ≤ 3

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Q. Solve the inequality: x^2 - 9 > 0.

A. x < -3 or x > 3
B. x > -3 and x < 3
C. x < 3
D. x > 3

Solution

Step 1: Factor: (x - 3)(x + 3) > 0. Step 2: Test intervals: solution is x < -3 or x > 3.

Correct Answer: A — x < -3 or x > 3

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Q. Solve the quadratic equation 3x^2 + 12x + 12 = 0.

A. x = -2
B. x = -4
C. x = 0
D. x = -6

Solution

Dividing the entire equation by 3 gives x^2 + 4x + 4 = 0, which factors to (x + 2)(x + 2) = 0. Thus, x = -2.

Correct Answer: B — x = -4

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Q. Solve the quadratic equation x^2 + 4x + 4 = 0.

A. -2
B. 2
C. 0
D. -4

Solution

This can be factored as (x + 2)(x + 2) = 0. Therefore, x = -2 is a double root.

Correct Answer: A — -2

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Q. Solve the quadratic equation x^2 + 6x + 9 = 0.

A. x = -3
B. x = 3
C. x = -6
D. x = 6

Solution

This can be factored as (x + 3)(x + 3) = 0. Therefore, x + 3 = 0 gives x = -3.

Correct Answer: A — x = -3

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Q. Solve the quadratic equation x^2 - 5x + 6 = 0.

A. x = 2, 3
B. x = 1, 6
C. x = -2, -3
D. x = 0, 5

Solution

Factor the equation: (x - 2)(x - 3) = 0. Thus, x = 2 or x = 3.

Correct Answer: A — x = 2, 3

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Q. Solve the quadratic inequality: x^2 - 5x + 6 < 0.

A. 1 < x < 6
B. 2 < x < 3
C. x < 2 or x > 3
D. 2 < x < 6

Solution

Step 1: Factor: (x - 2)(x - 3) < 0. Step 2: Test intervals: solution is 2 < x < 3.

Correct Answer: B — 2 < x < 3

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Q. Solve the system of equations: 2x + 3y = 6 and x - y = 1. What is the value of x?

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

Solution

From the second equation, x = y + 1. Substitute into the first equation: 2(y + 1) + 3y = 6. Simplifying gives 2y + 2 + 3y = 6, or 5y + 2 = 6. Thus, 5y = 4, so y = 0.8. Therefore, x = 0.8 + 1 = 1.8.

Correct Answer: B — 2

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Algebra (Secondary)

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