Inequalities and Their Applications

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Q1. 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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Q2. 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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Q3. 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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Q4. 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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Q5. What is the solution set for the inequality: 2x + 3 ≥ 11?

Solution:

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

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Q6. Find 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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Q7. Which of the following is a solution to 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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Q8. Which of the following values satisfies the inequality: x/3 + 2 < 5?

Solution:

Step 1: Subtract 2 from both sides: x/3 < 3. Step 2: Multiply by 3: x < 9.

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Q9. 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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Q10. If 3(x + 2) < 2(x + 5), what is the value of x?

Solution:

Step 1: Distribute: 3x + 6 < 2x + 10. Step 2: Subtract 2x from both sides: x + 6 < 10. Step 3: Subtract 6: x < 4.