Linear Programming Basics

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Q1. What is the solution set for the equation 4x + 1 = 2x + 9?

Solution:

Subtract 2x from both sides: 2x + 1 = 9. Then subtract 1: 2x = 8. Finally, divide by 2: x = 4.

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Q2. What is the solution set for the equation 4x + 8 = 0?

Solution:

Subtract 8 from both sides: 4x = -8. Then divide by 4: x = -2.

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Q3. Which of the following represents the factored form of x^2 + 5x + 6?

Solution:

Factor the polynomial: x^2 + 5x + 6 = (x + 2)(x + 3).

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Q4. What is the y-intercept of the line represented by the equation y = -3x + 7?

Solution:

The y-intercept is the constant term when x = 0, which is 7.

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Q5. What is the solution set for the inequality x + 2 > 5?

Solution:

Subtract 2 from both sides: x > 3.

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Q6. What is the value of x in the quadratic equation x^2 - 5x + 6 = 0?

Solution:

Factor the equation: (x - 2)(x - 3) = 0. Thus, x = 2 or x = 3.

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Q7. What is the factored form of x^2 - 9?

Solution:

This is a difference of squares: x^2 - 9 = (x - 3)(x + 3).

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Q8. What is the range of the function y = -x^2 + 4?

Solution:

The vertex is at (0, 4) and opens downwards, so the range is y ≤ 4.