Inequalities and Their Applications

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Q1. Which of the following values satisfies the inequality: 5 - 2x > 1?

Solution:

Step 1: Subtract 5 from both sides: -2x > -4. Step 2: Divide by -2 (reverse inequality): x < 2.

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Q2. What is the solution to the inequality: 3(x + 2) ≤ 2(x + 5)?

Solution:

Step 1: Distribute: 3x + 6 ≤ 2x + 10. Step 2: Subtract 2x: x + 6 ≤ 10. Step 3: Subtract 6: x ≤ 4.

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Q3. What is the solution to the inequality: 2(x - 1) > 3(x + 2)?

Solution:

Step 1: Distribute: 2x - 2 > 3x + 6. Step 2: Rearrange: -x > 8. Step 3: Divide by -1 (reverse inequality): x < -8.

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Q4. 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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Q5. 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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Q6. 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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Q7. 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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Q8. Solve the inequality: 4x - 7 > 2x + 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.

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Q9. 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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Q10. Solve the inequality: x^2 - 4 > 0.

Solution:

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