If the values of x and y are negative, which of the following values must be positive?
- A. x²-y²
- B. x/y
- C. x+y
- D. x-y
Correct Answer & Rationale
Correct Answer: B
When both x and y are negative, the quotient \( x/y \) results in a positive value. This is because dividing a negative number by another negative number yields a positive outcome. Option A, \( x^2 - y^2 \), can be either positive or negative depending on the magnitudes of x and y; thus, it is not guaranteed to be positive. Option C, \( x + y \), is the sum of two negative numbers, which will always be negative. Option D, \( x - y \), involves subtracting a negative (y) from another negative (x), which can also yield a negative or zero result, depending on their values. Only \( x/y \) is assuredly positive.
When both x and y are negative, the quotient \( x/y \) results in a positive value. This is because dividing a negative number by another negative number yields a positive outcome. Option A, \( x^2 - y^2 \), can be either positive or negative depending on the magnitudes of x and y; thus, it is not guaranteed to be positive. Option C, \( x + y \), is the sum of two negative numbers, which will always be negative. Option D, \( x - y \), involves subtracting a negative (y) from another negative (x), which can also yield a negative or zero result, depending on their values. Only \( x/y \) is assuredly positive.
Other Related Questions
For what values of x does 5x ^ 2 + 4x - 4 = 0 ?
- A. x = 1/5 and x = - 1
- B. x = - 4/5 and x = 1
- C. x = (- 2±6 * √(2))/5
- D. x = (- 2±2 * √(6))/5
Correct Answer & Rationale
Correct Answer: D
To solve the quadratic equation \(5x^2 + 4x - 4 = 0\), one can apply the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Here, \(a = 5\), \(b = 4\), and \(c = -4\). Calculating the discriminant gives \(b^2 - 4ac = 16 + 80 = 96\), leading to \(x = \frac{-4 \pm \sqrt{96}}{10} = \frac{-4 \pm 4\sqrt{6}}{10} = \frac{-2 \pm 2\sqrt{6}}{5}\), which matches option D. Option A provides incorrect roots not derived from the quadratic formula. Option B also presents incorrect values, failing to satisfy the equation. Option C miscalculates the discriminant, leading to an incorrect expression. Thus, D accurately reflects the solution to the equation.
To solve the quadratic equation \(5x^2 + 4x - 4 = 0\), one can apply the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Here, \(a = 5\), \(b = 4\), and \(c = -4\). Calculating the discriminant gives \(b^2 - 4ac = 16 + 80 = 96\), leading to \(x = \frac{-4 \pm \sqrt{96}}{10} = \frac{-4 \pm 4\sqrt{6}}{10} = \frac{-2 \pm 2\sqrt{6}}{5}\), which matches option D. Option A provides incorrect roots not derived from the quadratic formula. Option B also presents incorrect values, failing to satisfy the equation. Option C miscalculates the discriminant, leading to an incorrect expression. Thus, D accurately reflects the solution to the equation.
Which equation is a correct way to calculate x?
- A. sin x=5,000 /7,000
- B. sin x= 7,000 /5,000
- C. tan x= 5,000/7,000
- D. tan x=7,000/5,000
Correct Answer & Rationale
Correct Answer: C
To solve for \( x \), the correct relationship involves the tangent function, as \( \tan \) is defined as the ratio of the opposite side to the adjacent side in a right triangle. Option C, \( \tan x = \frac{5,000}{7,000} \), accurately represents this ratio. Option A misapplies the sine function, which should represent the ratio of the opposite side to the hypotenuse, not the adjacent side. Similarly, option B incorrectly uses sine but with the sides reversed, leading to an inaccurate representation. Option D misuses tangent, suggesting the opposite and adjacent sides are swapped, which does not align with the definition of tangent. Thus, only option C correctly applies the tangent function to find \( x \).
To solve for \( x \), the correct relationship involves the tangent function, as \( \tan \) is defined as the ratio of the opposite side to the adjacent side in a right triangle. Option C, \( \tan x = \frac{5,000}{7,000} \), accurately represents this ratio. Option A misapplies the sine function, which should represent the ratio of the opposite side to the hypotenuse, not the adjacent side. Similarly, option B incorrectly uses sine but with the sides reversed, leading to an inaccurate representation. Option D misuses tangent, suggesting the opposite and adjacent sides are swapped, which does not align with the definition of tangent. Thus, only option C correctly applies the tangent function to find \( x \).
If the function g is defined by g (x) = x/(x+1)', which of the following is true?
- A. g (10) <g (20)
- B. g (20) <g (10)
- C. g(0) =1
- D. g(1)=0
Correct Answer & Rationale
Correct Answer: A
To analyze the function \( g(x) = \frac{x}{x+1} \), we first observe its behavior as \( x \) increases. The function \( g(x) \) is a rational function that approaches 1 as \( x \) approaches infinity. For option A, evaluating \( g(10) \) and \( g(20) \): - \( g(10) = \frac{10}{11} \approx 0.909 \) - \( g(20) = \frac{20}{21} \approx 0.952 \) Since \( 0.909 < 0.952 \), option A is true. For option B, it incorrectly suggests \( g(20) < g(10) \), which contradicts the findings. Option C states \( g(0) = 1 \), but \( g(0) = 0 \), making this option false. Option D claims \( g(1) = 0 \), while \( g(1) = \frac{1}{2} \), which is also incorrect. Thus, only option A holds true.
To analyze the function \( g(x) = \frac{x}{x+1} \), we first observe its behavior as \( x \) increases. The function \( g(x) \) is a rational function that approaches 1 as \( x \) approaches infinity. For option A, evaluating \( g(10) \) and \( g(20) \): - \( g(10) = \frac{10}{11} \approx 0.909 \) - \( g(20) = \frac{20}{21} \approx 0.952 \) Since \( 0.909 < 0.952 \), option A is true. For option B, it incorrectly suggests \( g(20) < g(10) \), which contradicts the findings. Option C states \( g(0) = 1 \), but \( g(0) = 0 \), making this option false. Option D claims \( g(1) = 0 \), while \( g(1) = \frac{1}{2} \), which is also incorrect. Thus, only option A holds true.
How many cups of peanut butter must be used in order to make exactly enough peanut butter balls for the children at the party?
- A. 10
- B. 12
- C. 18
- D. 24
Correct Answer & Rationale
Correct Answer: C
To determine the number of cups of peanut butter needed for the peanut butter balls, one must consider the recipe's requirements and the number of children attending the party. Option C (18 cups) aligns with the recipe's proportion to yield the exact quantity necessary for all children. Option A (10 cups) is insufficient, likely resulting in fewer peanut butter balls than required. Option B (12 cups) may also fall short, leading to a shortage. Option D (24 cups) exceeds the needed amount, creating waste. Thus, C is the optimal choice, ensuring each child receives a peanut butter ball without excess or deficit.
To determine the number of cups of peanut butter needed for the peanut butter balls, one must consider the recipe's requirements and the number of children attending the party. Option C (18 cups) aligns with the recipe's proportion to yield the exact quantity necessary for all children. Option A (10 cups) is insufficient, likely resulting in fewer peanut butter balls than required. Option B (12 cups) may also fall short, leading to a shortage. Option D (24 cups) exceeds the needed amount, creating waste. Thus, C is the optimal choice, ensuring each child receives a peanut butter ball without excess or deficit.