answer:1. **Rewrite the inequality**: Start by rewriting the given inequality: [ 4x^3 + 9x^2 - 6x < 2 ] Subtract 2 from both sides to set the inequality to zero: [ 4x^3 + 9x^2 - 6x - 2 < 0 ] 2. **Factoring the cubic expression**: We attempt to factor the cubic expression: - The rational root theorem suggests possible rational roots as factors of the constant term (-2) over factors of the leading coefficient (4), i.e., ±1, ±1/2, ±2, and ±1/4. - By substituting these values, we find that (x = -1) is a root. - Use synthetic division to factor out (x + 1), yielding: [ 4x^3 + 9x^2 - 6x - 2 = (x + 1)(4x^2 + 5x - 2) ] - Further factorize (4x^2 + 5x - 2) using the middle term breaking method or quadratic formula: [ 4x^2 + 5x - 2 = (4x - 1)(x + 2) ] 3. **Analyze the critical points**: - The roots are (x = -1), (x = 1/4), and (x = -2). - Determine the sign of the expression ((x + 1)(4x - 1)(x + 2)) in the intervals determined by these roots. 4. **Determine the sign of the product in each interval**: - (x < -2): Choose (x = -3); calculation gives a positive sign. - (-2 < x < -1): Choose (x = -1.5); calculation gives a negative sign. - (-1 < x < 1/4): Choose (x = 0); calculation gives a negative sign. - (x > 1/4): Choose (x = 1); calculation gives a positive sign. 5. **Conclude the solution**: The expression ((x + 1)(4x - 1)(x + 2)) is less than zero for intervals (-2 < x < -1) and (-1 < x < 1/4). Thus, the solution to the inequality 4x^3 + 9x^2 - 6x < 2 is: [ -2 < x < 1/4 ] The final answer is boxed{textbf{(B) } -2 < x < -1 text{ or } -1 < x < 1/4}

answer:Solution: (1) 8^{ frac {2}{3}} = 4 = frac {32}{8}, ( frac {16}{81})^{- frac {3}{4}} = ( frac {2}{3})^{4 times (- frac {3}{4})} = frac {27}{8}, therefore 8^{ frac {2}{3}} > ( frac {16}{81})^{- frac {3}{4}}; (2) Since y = ln x is an increasing function, and 10 > e, therefore ln 10 > ln e; (3) Since y = 0.8^x is a decreasing function, therefore 0.8^{-0.1} < 0.8^{-0.2}; (4) Since y = x^{0.1} is an increasing function, therefore 8^{0.1} < 9^{0.1}. Hence, the number of correct conclusions is 2, Therefore, the answer is: boxed{B}. This question examines the application of the monotonicity of exponential, power, and logarithmic functions, and is considered a basic question.

answer:1. The original center of the circle is at (8, -3). 2. Reflecting a point over the line y=-x involves switching the coordinates and changing their signs. - If (x, y) is the original point, then after reflection about y=-x, the new coordinates will be (-y, -x). 3. Apply this reflection to the center point (8, -3): - Switch the coordinates and change their signs: (-(-3), -(8)) = (3, -8). So, the new coordinates of the center of the reflected circle are boxed{(3, -8)}.

answer:First, let's calculate the number of fish each person caught: Jordan caught 4 fish. Perry caught double what Jordan caught, so Perry caught 4 * 2 = 8 fish. The sum of what Jordan and Perry caught is 4 + 8 = 12 fish. Alex caught three times the sum of what Jordan and Perry caught, so Alex caught 12 * 3 = 36 fish. Now, let's find out the total number of fish they caught before the bird incident: Total fish = Jordan's fish + Perry's fish + Alex's fish Total fish = 4 + 8 + 36 Total fish = 48 fish After the bird stole 2 of their fish, they were left with: Total fish after bird theft = 48 - 2 Total fish after bird theft = 46 fish Finally, they decided to release one-third of the remaining fish back into the lake: Fish to release = 1/3 * 46 Fish to release = 15.333... (Since they can't release a fraction of a fish, we'll round down to the nearest whole number, which is 15 fish.) The number of fish remaining after releasing some back into the lake is: Fish remaining = Total fish after bird theft - Fish to release Fish remaining = 46 - 15 Fish remaining = 31 fish So, they were left with boxed{31} fish.