Inequalities and Their Applications

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Q1. 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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Q2. Find the solution to the inequality: 4x + 1 ≥ 2x + 5.

Solution:

Step 1: Subtract 2x from both sides: 2x + 1 ≥ 5. Step 2: Subtract 1: 2x ≥ 4. Step 3: Divide by 2: x ≥ 2.

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Q3. Find 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: x^2 + 2x - 8 < 0.

Solution:

Step 1: Factor: (x + 4)(x - 2) < 0. Step 2: Critical points are x = -4 and x = 2. Step 3: Test intervals: -4 < x < 2.

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Q5. 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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Q6. 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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Q7. 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.

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Q8. 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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Q9. Which of the following values satisfies the inequality: 5 - x > 2?

Solution:

Step 1: Subtract 5 from both sides: -x > -3. Step 2: Multiply by -1 (reverse the inequality): x < 3.

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Q10. What is the solution set for the inequality: -4x + 1 < 3?

Solution:

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