Inequalities and Their Applications

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Q1. Which of the following is a solution to the inequality: -4x + 1 ≤ 9?

Solution:

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

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Q2. If -x + 6 > 2, what is the solution for x?

Solution:

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

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Q3. If 3(x - 1) < 2x + 4, what is the value of x?

Solution:

Step 1: Distribute: 3x - 3 < 2x + 4. Step 2: Subtract 2x: x - 3 < 4. Step 3: Add 3: x < 7.

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Q4. Which of the following values satisfies the inequality: 4x - 1 < 3x + 2?

Solution:

Step 1: Subtract 3x from both sides: x - 1 < 2. Step 2: Add 1: x < 3.

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Q5. Which of the following values satisfies the inequality: 4 - x > 1?

Solution:

Step 1: Subtract 4 from both sides: -x > -3. Step 2: Multiply by -1 (reverse inequality): x < 3.

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Q6. Solve 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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Q7. What is the solution to the inequality: x^2 - 9 > 0?

Solution:

Step 1: Factor: (x - 3)(x + 3) > 0. Step 2: Test intervals: solution is x < -3 or x > 3.

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Q8. What is the solution to the inequality: 2(x - 1) < 3(x + 2)?

Solution:

Step 1: Distribute: 2x - 2 < 3x + 6. Step 2: Subtract 2x: -2 < x + 6. Step 3: Subtract 6: -8 < x, or x > -8.

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Q9. Solve the inequality: 7 - 2x ≥ 3.

Solution:

Step 1: Subtract 7 from both sides: -2x ≥ -4. Step 2: Divide by -2 (reverse inequality): x ≤ 2.

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Q10. Solve the inequality: 7x + 1 ≤ 3x + 9.

Solution:

Step 1: Subtract 3x from both sides: 4x + 1 ≤ 9. Step 2: Subtract 1: 4x ≤ 8. Step 3: Divide by 4: x ≤ 2.