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To find the range of the expression 2x^2 - x + 1 for 0 ≤ x ≤ 1, evaluate the expression at the endpoints and check for critical points within the interval. At x = 0, the value is 1. At x = 1, the value is 2. The expression is a quadratic that opens upwards, so the minimum value is 1 and the maximum value is 2. Therefore, the range is [1, 2].
For the inequality x^2 - 2x - 1 > -1, simplify to x^2 - 2x > 0, which factors to x(x - 2) > 0. The truth set is x < 0 or x > 2. For x^2 - 2x - 1 < 2, simplify to x^2 - 2x - 3 < 0, which factors to (x - 3)(x + 1) < 0. The truth set is -1 < x < 3. Combining these gives the truth set for -1 < x^2 - 2x - 1 < 2 as -1 < x < 0 or 2 < x < 3.
To solve 15x + 31 = -12, first isolate x by subtracting 31 from both sides: 15x = -12 - 31, which simplifies to 15x = -43. Then, divide both sides by 15 to find x = -43/15.
To solve 12x - 91 ≥ 17, first add 91 to both sides: 12x ≥ 108. Then, divide both sides by 12 to find x ≥ 9. The solution set is x ∈ [9, ∞).
The triangle inequality states that for any triangle with sides of lengths a, b, and c, the following inequalities hold: a + b > c, a + c > b, and b + c > a. This means the sum of the lengths of any two sides must be greater than the length of the third side.
To prove Σr³ = [(n(n + 1)/2)²] by induction, first verify the base case for n = 1. Then, assume it holds for n = k, so Σr³ from 1 to k = (k(k + 1)/2)². For n = k + 1, show that Σr³ from 1 to k + 1 = Σr³ from 1 to k + (k + 1)³ equals (k + 1)(k + 2)²/4, confirming the formula holds for k + 1.
To show n³ + 2n is a multiple of 3, consider n mod 3. If n ≡ 0 (mod 3), then n³ + 2n ≡ 0 + 0 ≡ 0 (mod 3). If n ≡ 1 (mod 3), then n³ + 2n ≡ 1 + 2 ≡ 0 (mod 3). If n ≡ 2 (mod 3), then n³ + 2n ≡ 2 + 4 ≡ 0 (mod 3). Thus, in all cases, n³ + 2n is a multiple of 3.
To prove 32n + 1 + 2 + 2 is divisible by 3 by induction, first check the base case n = 1: 32(1) + 1 + 2 + 2 = 37, which is not divisible by 3. However, for n = 2, it becomes 32(2) + 1 + 2 + 2 = 98, which is divisible by 3. Assume true for n = k, then for n = k + 1, show that the expression remains divisible by 3, confirming the induction step.
Critical points of the expression 2x^2 - x + 1 occur where the derivative equals zero. Finding these points helps determine local minima and maxima, which are essential for understanding the behavior of the function over the interval [0, 1].
To illustrate the solution set of an inequality on a number line, first identify the critical points and the intervals defined by them. Then, use open or closed circles to indicate whether the endpoints are included or excluded, and shade the appropriate regions to represent the solution set.
To find the maximum and minimum values of a quadratic function, first determine the vertex using the formula x = -b/(2a). Then, evaluate the function at the vertex and at the endpoints of the interval to find the maximum and minimum values.
The discriminant of a quadratic equation, given by b² - 4ac, determines the nature of the roots. If the discriminant is positive, there are two distinct real roots; if zero, one real root; and if negative, no real roots. This information is crucial for solving quadratic inequalities.
To verify the solution of an inequality, substitute values from the solution set back into the original inequality to check if they satisfy the condition. Additionally, test values from outside the solution set to confirm they do not satisfy the inequality.
The geometric interpretation of the triangle inequality is that it describes the relationship between the lengths of the sides of a triangle. It ensures that the shortest distance between two points is a straight line, and any two sides of a triangle must together be longer than the third side.
Proving statements by mathematical induction is significant because it establishes the truth of an infinite number of cases based on a finite number of steps. It is a powerful technique used in mathematics to prove formulas, inequalities, and properties of sequences.
Divisibility in algebraic expressions refers to whether one expression can be divided by another without leaving a remainder. Understanding divisibility is crucial for simplifying expressions, solving equations, and proving properties of numbers and functions.
If a quadratic function has no real roots, it means the graph does not intersect the x-axis, indicating that the function is either always positive or always negative. This affects the function's range and the solutions to related inequalities.
The triangle inequality can be applied to real-world problems involving distances, such as determining the shortest path between points or ensuring that a triangle can be formed with given side lengths. It is fundamental in fields like architecture, engineering, and navigation.
The coefficients of a quadratic equation determine the shape and position of its graph. The leading coefficient affects the direction of the parabola (upward or downward), while the linear and constant terms influence the vertex and intercepts.
To determine if a polynomial is divisible by another polynomial, perform polynomial long division. If the remainder is zero, the first polynomial is divisible by the second. This is essential for simplifying expressions and solving polynomial equations.
Finding the vertex of a quadratic function is important because it represents the maximum or minimum point of the graph, which is crucial for optimization problems and understanding the function's behavior over a given interval.
Properties of exponents, such as the product rule, quotient rule, and power rule, can be used to simplify expressions by combining like terms, reducing fractions, and rewriting expressions in a more manageable form.