Inequalities and Their Applications

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Q1. 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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Q2. Which of the following is a solution to 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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Q3. Solve 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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Q4. What is the solution to the inequality: 2(x - 1) ≥ 3x + 4?

Solution:

Step 1: Distribute: 2x - 2 ≥ 3x + 4. Step 2: Subtract 2x from both sides: -2 ≥ x + 4. Step 3: Subtract 4: -6 ≥ x. Solution: x ≤ -6.

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Q5. What is the solution to 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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Q6. Which of the following is a solution to the inequality: 2(x - 3) > 4?

Solution:

Step 1: Divide by 2: x - 3 > 2. Step 2: Add 3 to both sides: x > 5. Thus, x = 5 is a solution.

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Q7. 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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Q8. 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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Q9. Solve the inequality: 2(x - 1) ≥ 3x + 4.

Solution:

Step 1: Distribute: 2x - 2 ≥ 3x + 4. Step 2: Subtract 2x from both sides: -2 ≥ x + 4. Step 3: Subtract 4: -6 ≥ x. Step 4: Reverse: x ≤ -6.

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Q10. Find 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.