Inequalities and Their Applications

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Q1. Solve the inequality: 7 - 2x < 1.

Solution:

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

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Q2. What is the solution to the inequality: 4x + 1 ≥ 2x + 9?

Solution:

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

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Q3. Which of the following represents the solution to the inequality: 4x + 1 ≥ 2x + 9?

Solution:

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

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Q4. What is the solution set for the inequality: x^2 - 5x + 6 < 0?

Solution:

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

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Q5. What is the solution to the inequality: 3(x - 2) > 2(x + 1)?

Solution:

Step 1: Distribute: 3x - 6 > 2x + 2. Step 2: Subtract 2x: x - 6 > 2. Step 3: Add 6: x > 8.

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Q6. Which of the following is a solution to the inequality: 3x + 2 < 11?

Solution:

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

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Q7. Find the solution set for the inequality: x^2 - 4 > 0.

Solution:

Step 1: Factor the inequality: (x - 2)(x + 2) > 0. Step 2: The solution set is (-∞, -2) ∪ (2, ∞).

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Q8. Which of the following is a solution to the inequality: 6 - 2x ≤ 4?

Solution:

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

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Q9. Solve the inequality: x^2 - 5x + 6 ≤ 0.

Solution:

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

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Q10. Solve the inequality: x^2 + 2x - 8 < 0.

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.