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To solve the problem where two-thirds of a number reduced by 11 is equal to 4 more than the number, let’s define the number as x. According to the problem, we can form the following equation: Two-thirds of x is represented as ⅔x. Reducing it by 11 gives us: ⅔x – 11. 4 more than … Read more

To find the other factor of the polynomial 6x² + 15, given that one factor is 2x + 3, we will use polynomial division. First, let’s verify if 2x + 3 is indeed a factor of the polynomial 6x² + 15. We will do this by performing polynomial long division. Step 1: Divide the leading … Read more

To find the equation of the line that contains the points (0, 4) and (3, 2), we can use the slope-intercept form of a linear equation, which is y = mx + b, where m is the slope and b is the y-intercept. Step 1: Calculate the slope The slope m of a line through … Read more

To determine the range of the function f(x) = x^5 + x – 1, we start by analyzing its behavior as x approaches positive and negative infinity. As x approaches infinity, the term x^5, which is the leading term, dominates. Therefore, f(x) approaches infinity: f(x) → ∞ as x → ∞ Conversely, as x approaches … Read more

To determine the equation of a line that passes through the point (2, 1) with an undefined slope, we need to understand what an undefined slope represents. An undefined slope occurs when a line is vertical. This means that all points on the line have the same x-coordinate. In this scenario, since the line passes … Read more

To find the equations of the tangent lines to the curve y = x2 + 1 that are parallel to the line 2y = 5, we need to follow a few steps: Determine the slope of the given line. First, we need to rewrite the equation of the line 2y = 5 in slope-intercept form … Read more

To identify which graph represents the function f(x) = x3 + x2 + 10x + 8, we need to analyze the characteristics of the function, such as its shape, intercepts, and behavior at infinity. This function is a polynomial of degree 3, meaning its leading coefficient and degree will dictate its end behavior. As x … Read more

To find the length of the shorter leg in a 30-60-90 triangle when the hypotenuse is 30, we can use the properties of this special triangle. A 30-60-90 triangle has side lengths that are in a fixed ratio: the length of the shorter leg (opposite the 30-degree angle) is half the length of the hypotenuse, … Read more

The given sequence is: 3, 12, 48, 192. To find the equation for the nth term, we first examine how the sequence is generated. We can observe the following pattern: The first term (n=1) is 3. The second term (n=2) is 12, which is 3 x 4. The third term (n=3) is 48, which is … Read more

To determine how many glasses of 25 ml can be filled from a vessel containing 4 liters and 500 ml of curd, we first need to convert everything into the same unit. A liter is equal to 1000 ml, so: 4 liters = 4 x 1000 ml = 4000 ml Adding the additional 500 ml … Read more