Indices and Surds - Applications

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Q1. Which of the following is a solution to the inequality 5x - 7 < 8?

Solution:

Add 7 to both sides: 5x < 15. Divide by 5: x < 3.

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Q2. What is the value of x in the polynomial equation x^3 - 4x = 0?

Solution:

Factor out x: x(x^2 - 4) = 0. Thus, x = 0 or x^2 - 4 = 0, giving x = 2 or x = -2.

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Q3. Which expression represents the sum of the roots of the equation x^2 - 6x + 8 = 0?

Solution:

Step 1: The sum of the roots is given by -b/a: 6/1 = 6.

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Q4. What is the solution to the inequality 2x + 1 > 7?

Solution:

Subtract 1 from both sides: 2x > 6. Then divide by 2: x > 3.

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Q5. What is the value of x in the equation 3(x - 2) = 12?

Solution:

Divide both sides by 3: x - 2 = 4. Then add 2: x = 6.

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Q6. If 2x + 3y = 12 and y = 2, what is the value of x?

Solution:

Step 1: Substitute y = 2 into the equation: 2x + 6 = 12. Step 2: Subtract 6: 2x = 6. Step 3: Divide by 2: x = 3.

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Q7. What is the solution to the equation 4(x - 1) = 12?

Solution:

Divide both sides by 4: x - 1 = 3. Add 1 to both sides: x = 4.

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Q8. Solve for x in the equation x^2 + 6x + 9 = 0.

Solution:

Factor the equation: (x + 3)(x + 3) = 0. Thus, x = -3.

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Q9. Which of the following is a solution to the inequality 3x - 5 < 4?

Solution:

Add 5 to both sides: 3x < 9. Then divide by 3: x < 3.

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Q10. If 5x + 2 > 12, what is the smallest integer value of x?

Solution:

Subtract 2 from both sides: 5x > 10. Then divide by 5: x > 2.