Inequalities and Their Applications

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Q1. What is the solution to the inequality: 4x - 1 < 3x + 2?

Solution:

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

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Q2. Determine the solution set for the inequality: 2x^2 - 8 < 0.

Solution:

Step 1: Factor the inequality: 2(x^2 - 4) < 0. Step 2: Roots are x = -2 and x = 2. Step 3: The solution is between the roots: (-2, 2).

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Q3. Find the solution set for the inequality: 4x - 7 ≤ 9.

Solution:

Step 1: Add 7 to both sides: 4x ≤ 16. Step 2: Divide by 4: x ≤ 4.

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Q4. What is the solution to the inequality: 2x^2 - 8 < 0?

Solution:

Step 1: Factor: 2(x^2 - 4) < 0. Step 2: Further factor: 2(x - 2)(x + 2) < 0. Step 3: The solution is (-2, 2).

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Q5. Determine the solution for the inequality: 3(x - 1) < 2(x + 2).

Solution:

Step 1: Expand: 3x - 3 < 2x + 4. Step 2: Subtract 2x: x - 3 < 4. Step 3: Add 3: x < 7.

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Q6. Solve the inequality: 7x + 2 ≤ 3x + 10.

Solution:

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

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Q7. Find the solution to the inequality: x^2 - 9 > 0.

Solution:

Step 1: Factor the inequality: (x - 3)(x + 3) > 0. Step 2: The critical points are x = -3 and x = 3. Step 3: Test intervals: The solution set is (-∞, -3) ∪ (3, ∞).

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Q8. What is the solution to the inequality: -3x + 6 ≤ 0?

Solution:

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

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Q9. Find the solution set for the inequality: x^2 - 6x + 8 > 0.

Solution:

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

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Q10. What is the solution to the inequality: 2(x - 3) ≥ 4?

Solution:

Step 1: Distribute: 2x - 6 ≥ 4. Step 2: Add 6: 2x ≥ 10. Step 3: Divide by 2: x ≥ 5.

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