Inequalities and Their Applications

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Q1. Which of the following is a solution to the inequality: 2x + 3 > 11?

Solution:

Step 1: Subtract 3 from both sides: 2x > 8. Step 2: Divide by 2: x > 4.

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Q2. Solve the inequality: 7 - 3x < 1.

Solution:

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

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Q3. What is the solution to the inequality: 6 - x < 2?

Solution:

Step 1: Subtract 6 from both sides: -x < -4. Step 2: Multiply by -1 (reverse the inequality): x > 4.

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Q4. What is the solution set for the inequality: -4x + 1 ≤ 9?

Solution:

Step 1: Subtract 1 from both sides: -4x ≤ 8. Step 2: Divide by -4 (reverse the inequality): x ≥ -2.

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Q5. What is the solution to the inequality: 3x + 4 ≤ 2x + 10?

Solution:

Step 1: Subtract 2x from both sides: x + 4 ≤ 10. Step 2: Subtract 4 from both sides: x ≤ 6.

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Q6. Find the solution set for the inequality: 3x + 2 > 5.

Solution:

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

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Q7. What is the solution to the inequality: x^2 + 3x < 10?

Solution:

Step 1: Rearrange: x^2 + 3x - 10 < 0. Step 2: Factor: (x + 5)(x - 2) < 0. Step 3: The solution is -5 < x < 2.

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Q8. Which of the following values satisfies 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. Thus, x = 2 satisfies the inequality.

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Q9. Find the solution to the inequality: 4x - 7 ≥ 5.

Solution:

Step 1: Add 7 to both sides: 4x ≥ 12. Step 2: Divide by 4: x ≥ 3.

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Q10. What is the solution to the inequality: x^2 - 5x + 6 < 0?

Solution:

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