Inequalities and Their Applications

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Q1. What is the solution to the inequality: 2(x - 3) > 4?

Solution:

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

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Q2. 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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Q3. Which of the following represents the solution to the inequality: x^2 - 9 > 0?

Solution:

Step 1: Factor the inequality: (x - 3)(x + 3) > 0. Step 2: The solution is outside the roots: (-∞, -3) ∪ (3, ∞).

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Q4. What is the solution to the inequality: x^2 + 4x < 5?

Solution:

Step 1: Rearrange: x^2 + 4x - 5 < 0. Step 2: Factor: (x + 5)(x - 1) < 0. Step 3: Solution is -5 < x < 1.

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Q5. 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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Q6. Find the solution set for the inequality: x^2 + 2x - 8 > 0.

Solution:

Step 1: Factor the quadratic: (x - 2)(x + 4) > 0. Step 2: The solution is outside the roots: (-∞, -4) ∪ (2, ∞).

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Q7. 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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Q8. Determine the solution set for the inequality: x^2 - 6x + 8 ≤ 0.

Solution:

Step 1: Factor: (x - 2)(x - 4) ≤ 0. Step 2: The solution is between the roots: [2, 4].

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Q9. 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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Q10. If 4x - 7 < 9, what is the maximum value of x?

Solution:

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