answer:Given the function ( f(x) = -x^2 + x + m + 2 ). We need to determine the range of real values for ( m ) such that the inequality ( f(x) geq |x| ) has exactly one integer solution. 1. **Rewrite the inequality**: [ f(x) geq |x| Rightarrow -x^2 + x + m + 2 geq |x| ] 2. **Consider two parts of the function**: Define: [ g(x) = 2 - |x| quad text{and} quad h(x) = x^2 - x - m ] The inequality becomes: [ g(x) geq h(x) Rightarrow 2 - |x| geq x^2 - x - m ] 3. **Find the intersection**: Identify ( x ) where the integer solution can only be ((x = 0)), given both functions need to be equal at ( x = 0 ): [ f(0) = m + 2 quad text{and} quad g(0) = 2 Rightarrow m + 2 geq 0 Rightarrow m geq -2 ] 4. **Check for larger or smaller ( x )**: We need to ensure that no other integer value of ( x ) satisfies the inequality. Particularly for ( x = 1 ): [ f(1) = 2 + m - 1 quad text{and} quad |1| = 1 Rightarrow 2 + m - 1 < 1 Rightarrow 1 + m < 1 Rightarrow m < -1 ] 5. **Combine the results**: From the above step, collate the conditions: [ -2 leq m < -1 ] # Conclusion: The range of real values for ( m ) is: [ boxed{[-2, -1)} ]

answer:This problem requires the use of cofunction identities and trigonometric quotient identities. First, apply the cofunction identities to simplify the expression in the numerator. Then, use the trigonometric quotient identity to convert the expression into a tangent function. Step 1: Apply cofunction identities: - sin(pi - alpha) = sin alpha - sin(frac{pi}{2} + alpha) = cos alpha Step 2: Replace the expressions in the numerator: frac{sin alpha - cos alpha}{cos (-alpha)} Step 3: Use the trigonometric identity cos(-alpha) = cos alpha: frac{sin alpha - cos alpha}{cos alpha} Step 4: Separate the terms in the numerator: frac{sin alpha}{cos alpha} - frac{cos alpha}{cos alpha} Step 5: Apply the trigonometric quotient identity frac{sin alpha}{cos alpha} = tan alpha: tan alpha - 1 Step 6: Given that tan alpha = 4, substitute the value into the expression: 4 - 1 = boxed{3}

answer:1. **Calculate 25% less than 80**: To find 25% less than a number, we subtract 25% of the number from the number itself. [ 80 - 0.25 times 80 = 80 - 20 = 60 ] Alternatively, we can calculate this as 75% of 80: [ 0.75 times 80 = 60 ] 2. **Set up the equation for one-fourth more than a number**: Let n be the number we are trying to find, then one-fourth more than n is n plus one-fourth of n: [ n + frac{1}{4}n = frac{5}{4}n ] We know from step 1 that this expression equals 60: [ frac{5}{4}n = 60 ] 3. **Solve for n**: To find n, multiply both sides of the equation by the reciprocal of frac{5}{4}, which is frac{4}{5}: [ n = frac{4}{5} times 60 = 48 ] 4. **Conclusion**: The number that, when increased by 25%, equals 60 is 48. Therefore, the correct answer is 48. The final answer is boxed{C}.

answer:1. **Requirement**: Linda wants her trains arranged in rows of exactly 8 each. Thus, the total number of trains must be a multiple of 8. 2. **Current Count of Trains**: She currently owns 31 trains. 3. **Next Multiple of 8**: We seek the smallest multiple of 8 that is greater than or equal to 31. The multiples of 8 are ... 16, 24, 32, 40, etc. The smallest multiple of 8 greater than 31 is 32. 4. **Calculate Additional Trains Needed**: [ text{Additional trains} = 32 - 31 = 1 ] 5. **Conclusion**: To achieve the arrangement of 8 trains per row, Linda needs to buy 1 more train, leading us to the conclusion that the smallest number of additional trains required is 1. The final answer is The correct choice is boxed{textbf{A}}.