💡First, I need to find the total sum of all the integers from -12 to 12 inclusive. This can be done by recognizing that the series is symmetric around zero. For every positive integer, there is a corresponding negative integer of the same magnitude. Therefore, the sum of all these integers will be zero.Next, since the integers are arranged in a 5x5 grid, there are 5 rows and 5 columns. The total sum of all the numbers in the grid is zero, as calculated earlier. To find the common sum for each row and each column, I divide the total sum by the number of rows (or columns). Dividing zero by five gives zero. Therefore, each row and each column must sum to zero.

💡Alright, so I have this problem here about transforming the graph of y = sin(x). The instructions say to move all the points on the graph to the left by π/3 units and then extend the horizontal coordinates of all the points to twice their original length, keeping the vertical coordinates the same. I need to figure out what the equation of the resulting graph is, and then choose the correct option from A to D.First, I remember that when you move a graph left or right, it's a horizontal shift. For sine functions, shifting left by π/3 would mean replacing x with (x + π/3) in the equation. So, after the shift, the equation becomes y = sin(x + π/3). That makes sense because adding inside the function argument shifts it to the left.Next, the problem says to extend the horizontal coordinates to twice their original length. I think this means stretching the graph horizontally by a factor of 2. In function transformations, stretching horizontally by a factor of 2 is done by replacing x with x/2 in the equation. So, if I apply this to the already shifted function, it should be y = sin((x/2) + π/3). Wait, is that right?Let me double-check. If I have y = sin(x + π/3) and I want to stretch it horizontally by a factor of 2, I need to replace x with x/2. So, substituting, it becomes y = sin((x/2) + π/3). Hmm, that seems correct. But let me think about the order of operations here. Is the shift done before the stretch or after?Actually, in function transformations, the order matters. If you shift first and then stretch, the shift is applied before the stretching. So, the shift is inside the function, and then the stretching affects the entire input. So, if I first shift left by π/3, getting y = sin(x + π/3), and then stretch horizontally by 2, it's y = sin((x/2) + π/3). That seems correct.But wait, another way to think about it is that stretching affects the period of the sine function. The period of sin(x) is 2π. If I stretch it horizontally by 2, the period becomes 4π. So, the equation should have a coefficient of 1/2 in front of x to achieve that. So, y = sin((1/2)x + π/3). That matches what I had earlier.Looking at the options, option A is y = sin(x/2 + π/3), which is the same as y = sin((x/2) + π/3). So, that would be the correct one, right?But hold on, let me make sure I didn't mix up the order. Sometimes, when you have multiple transformations, the order can affect the result. For example, if you first stretch and then shift, it's different from shifting and then stretching. In this problem, the instructions say to first shift left and then stretch horizontally. So, I think my initial approach is correct.Let me visualize it. If I have the graph of y = sin(x), shifting it left by π/3 moves every point π/3 units to the left. Then, stretching it horizontally by a factor of 2 would make the graph wider, stretching each x-coordinate to twice its original length. So, the period becomes twice as long, which is why the coefficient of x is 1/2.Alternatively, if I had stretched first, the equation would be y = sin((x/2)), and then shifting left by π/3 would result in y = sin((x/2) + π/3). So, either way, the result is the same. Interesting.Wait, is that true? Let me think. If I first stretch y = sin(x) to y = sin(x/2), which has a period of 4π, and then shift left by π/3, it becomes y = sin((x/2) + π/3). On the other hand, if I first shift y = sin(x) to y = sin(x + π/3), and then stretch, it's also y = sin((x/2) + π/3). So, in this case, the order doesn't matter because both transformations are applied to the x inside the function, and they both affect the argument of the sine function.But actually, in general, the order does matter when you have both horizontal shifts and stretches. For example, if you have y = sin(2x + π/3), that's different from y = sin(2(x + π/3)) = sin(2x + 2π/3). So, in that case, the order affects the phase shift.But in our problem, since we're stretching after shifting, it's equivalent to y = sin((x/2) + π/3). So, that's why the coefficient of x is 1/2 and the phase shift is π/3.Looking back at the options, A is y = sin(x/2 + π/3), which is exactly what we have. So, that should be the correct answer.Wait, but let me check the other options just to be thorough. Option B is y = sin(x/2 + π/6). That would be if the phase shift was only π/6, which is half of π/3. Maybe someone might think that stretching affects the phase shift as well, but in reality, the phase shift is added after the stretching. So, if you first shift and then stretch, the phase shift remains π/3, not π/6.Option C is y = sin(2x + π/3). That would be if you compressed the graph horizontally by a factor of 2 and then shifted it left by π/3. But we're supposed to stretch, not compress, so this is incorrect.Option D is y = sin(2x - π/3). That would be a compression by factor 2 and a shift to the right by π/3, which is the opposite of what we need. So, definitely not.Therefore, after carefully considering the transformations and verifying the order, I conclude that option A is the correct answer.**Final Answer**boxed{A}

💡First, I need to calculate the average speed for each participant in the race. Since all participants took the same amount of time, which is 8 hours, I can use the formula:[text{Average Speed} = frac{text{Distance}}{text{Time}}]Starting with Cyclist A:[text{Cyclist A's average speed} = frac{100 text{ miles}}{8 text{ hours}} = 12.5 text{ mph}]Next, Cyclist B:[text{Cyclist B's average speed} = frac{55 text{ miles}}{8 text{ hours}} = 6.875 text{ mph}]For Car X:[text{Car X's average speed} = frac{160 text{ miles}}{8 text{ hours}} = 20 text{ mph}]For Car Y:[text{Car Y's average speed} = frac{120 text{ miles}}{8 text{ hours}} = 15 text{ mph}]And finally, Car Z:[text{Car Z's average speed} = frac{200 text{ miles}}{8 text{ hours}} = 25 text{ mph}]Now, I identify the highest and lowest average speeds:- **Highest average speed:** Car Z at 25 mph- **Lowest average speed:** Cyclist B at 6.875 mphTo find the difference between the highest and lowest average speeds:[text{Difference} = 25 text{ mph} - 6.875 text{ mph} = 18.125 text{ mph}]

💡Okay, so I have this geometry problem here, and I need to figure out the length of CE. Let me try to visualize the problem first. There's a triangle ABC, and the angle bisector of angle ABC is drawn. This bisector extends to meet the circumcircle of triangle ABC at point F. So, F is a point on the circumcircle, and it's along the angle bisector of angle B.Given that FB is 2 units long and EF is 1 unit, I need to find the length of CE. Hmm, okay. Let me try to draw this in my mind. Triangle ABC with circumcircle passing through A, B, and C. The angle bisector of angle B meets the circumcircle again at F. So, F is opposite to B in some way, but not necessarily the midpoint or anything like that.Since FB is 2 and EF is 1, I wonder where E is located. Wait, the problem mentions CE, so E must be a point on the circle as well, right? Or maybe E is somewhere else. Hmm, actually, the problem doesn't specify where E is, but since it's asking about CE, I think E must be another intersection point or perhaps a midpoint. Wait, maybe E is the midpoint of FC? Because EF is 1 and FC would be FE + EC, but I'm not sure yet.Let me think about the properties of angle bisectors in circles. I remember that the angle bisector theorem might come into play here. The angle bisector theorem states that the angle bisector divides the opposite side into segments proportional to the adjacent sides. But in this case, since we're dealing with a circle, maybe there's a different theorem or property that applies.Oh, right! There's a theorem that says if two chords intersect, the products of the segments are equal. But I'm not sure if that applies here. Alternatively, maybe power of a point could be useful. The power of a point with respect to a circle relates the lengths of segments created by lines through that point intersecting the circle.Wait, since F is on the circumcircle and the angle bisector, maybe there's some similar triangles involved. Let me try to recall. If two angles subtend the same arc, they are equal. So, maybe angles at F and C are related.Let me try to write down what I know:1. Triangle ABC with circumcircle passing through A, B, C.2. Angle bisector of angle B meets the circumcircle again at F.3. FB = 2, EF = 1.4. Need to find CE.I think I need to find some relationship between these segments. Maybe using similar triangles or power of a point.Since F is on the circumcircle, maybe there are some similar triangles involving F. Let me see. If I can find two triangles that are similar, I can set up a proportion.Wait, the angle bisector of angle B meets the circumcircle at F, so maybe triangle FBC is similar to some other triangle. Let me think about the angles.Since FB is the angle bisector, angle ABF is equal to angle FBC. Also, because F is on the circumcircle, angle AFB is equal to angle ACB because they subtend the same arc AB. Hmm, that might be useful.Alternatively, maybe I can use the Angle Bisector Theorem in triangle ABC. The theorem states that the angle bisector divides the opposite side into segments proportional to the adjacent sides. So, if the angle bisector meets AC at some point, say D, then AB/BC = AD/DC. But in this case, the angle bisector is extended to meet the circumcircle at F, so maybe there's a similar proportion involving FB and FC.Wait, I think there's a theorem that relates the lengths of segments created by an angle bisector extended to the circumcircle. Let me try to recall. I think it's something like FB * FC = something involving the sides of the triangle. Hmm, maybe not exactly, but perhaps similar.Alternatively, maybe I can use power of a point from point E. If E is a point outside the circle, then the power of E with respect to the circle is equal to the product of the lengths of the segments from E to the circle. But I'm not sure where E is located.Wait, the problem mentions EF = 1. So, E is a point such that EF is 1. If F is on the circumcircle, then E must be somewhere along the line from F, but I'm not sure if it's inside or outside the circle.Wait, maybe E is the midpoint of FC? If EF is 1, then FC would be 2, but then CE would be 1 as well, but that might not necessarily be the case. Hmm.Alternatively, maybe E is another intersection point of some line with the circle. But the problem doesn't specify, so I'm a bit confused.Wait, let me go back to the problem statement. It says, "the angle bisector of angle ABC in triangle ABC extends to meet the circumcircle at point F." So, F is the second intersection point of the angle bisector with the circumcircle. Then, it says FB = 2 and EF = 1. So, E must be a point on the angle bisector such that from E, the segment EF is 1, and from F, the segment FB is 2.So, if I imagine the angle bisector starting at B, going through some point E, and then extending to F on the circumcircle, with FB = 2 and EF = 1. So, the total length from E to F is 1, and from F to B is 2. So, the entire length from E to B would be EF + FB = 1 + 2 = 3.Wait, but E is on the angle bisector, so maybe E is between B and F? Or is E beyond F? Hmm, the problem doesn't specify, but since EF = 1 and FB = 2, it's more likely that E is between B and F, making the total length from E to F as 1, and from F to B as 2. So, E is closer to B than F is.But I'm not entirely sure. Maybe E is beyond F? Then, from E to F is 1, and from F to B is 2, so from E to B would be 3. But without more information, it's hard to tell.Wait, maybe I can use the power of point E with respect to the circumcircle. The power of E would be equal to EB * EF = EC * something. Hmm, but I'm not sure what that something is.Alternatively, maybe I can use similar triangles. Since F is on the circumcircle, and the angle bisector passes through F, maybe there are some similar triangles involving F and E.Wait, let me think about the angles. Since FB is the angle bisector, angle ABF is equal to angle FBC. Also, because F is on the circumcircle, angle AFB is equal to angle ACB, as they subtend the same arc AB.Similarly, angle AFC is equal to angle ABC, because they subtend the same arc AC.Hmm, maybe I can set up some proportions using these angle equalities.Alternatively, maybe I can use the theorem that states that if two chords intersect, the products of their segments are equal. But I'm not sure if that applies here.Wait, another thought: since F is on the circumcircle and on the angle bisector, maybe there's a relationship between the lengths FB and FC. I think there's a theorem that says that FB * FC = something involving the sides of the triangle, but I can't recall exactly.Wait, maybe I can use the Angle Bisector Theorem in triangle ABC. The theorem states that the angle bisector divides the opposite side into segments proportional to the adjacent sides. So, if the angle bisector meets AC at D, then AB/BC = AD/DC. But in this case, the angle bisector is extended to meet the circumcircle at F, so maybe there's a similar proportion involving FB and FC.Wait, I think there's a theorem that says that if an angle bisector is extended to meet the circumcircle, then the ratio of the lengths from the vertex to the points of intersection is equal to the ratio of the adjacent sides. So, maybe FB/FC = AB/BC.But I'm not sure if that's exactly correct. Let me try to recall.Alternatively, maybe I can use the theorem that says that FB * FC = AB * BC. Hmm, that might not be correct either.Wait, maybe I can use power of a point from point E. If E is on the angle bisector, and EF = 1, FB = 2, then the power of E with respect to the circumcircle is equal to EB * EF = EC * something.Wait, but I don't know where E is located exactly. Is E inside the circle or outside? If E is between B and F, then it's inside the circle, and the power of E would be negative. If E is outside, then it's positive.Wait, let me assume that E is inside the circle, so between B and F. Then, the power of E would be equal to EB * EF = EC * something. But I'm not sure what that something is.Alternatively, maybe I can consider triangle EFB and triangle EFC. If I can find some similarity or proportionality between them, I might be able to find CE.Wait, another idea: since F is on the circumcircle, and FB is the angle bisector, maybe triangle FBC is similar to triangle FBE. Let me check.If triangle FBC is similar to triangle FBE, then the corresponding angles would be equal. Let's see:- Angle FBC is equal to angle FBE because FB is the angle bisector, so angle FBC = angle FBE.- Also, angle BFC is equal to angle BEF because they subtend the same arc BC.Wait, is that correct? Hmm, angle BFC subtends arc BC, and angle BEF subtends arc BF. Hmm, maybe not.Wait, perhaps angle BFC is equal to angle BEF because of some cyclic quadrilateral properties. Hmm, I'm not sure.Alternatively, maybe triangle FBE is similar to triangle FCB. Let me see:- Angle FBE is equal to angle FCB because they both subtend arc FC.Wait, angle FBE is equal to angle FCB because they both subtend arc FC. So, angle FBE = angle FCB.Also, angle FEB is equal to angle FBC because they both subtend arc FC. Wait, no, angle FEB subtends arc FB, and angle FBC subtends arc FC. Hmm, maybe not.Wait, I'm getting confused. Let me try to write down the angles.In triangle FBC:- Angle at B: angle FBC.- Angle at C: angle FCB.- Angle at F: angle BFC.In triangle FBE:- Angle at B: angle FBE.- Angle at E: angle FEB.- Angle at F: angle BFE.Wait, maybe angle FBE is equal to angle FCB, as they both subtend arc FC.Similarly, angle FEB is equal to angle FBC, as they both subtend arc FB.Wait, if that's the case, then triangle FBE is similar to triangle FCB by AA similarity.Yes, that seems right. So, triangle FBE ~ triangle FCB.Therefore, the ratio of corresponding sides is equal.So, FB / FC = FE / FB.Given that FB = 2 and FE = 1, we can set up the proportion:FB / FC = FE / FB2 / FC = 1 / 2Cross-multiplying, we get:2 * 2 = FC * 1So, FC = 4.Therefore, FC = 4.Now, since FC = 4 and FE = 1, then CE = FC - FE = 4 - 1 = 3.So, CE = 3.Therefore, the answer is A: 3.Wait, let me double-check my reasoning.I assumed that triangle FBE is similar to triangle FCB because they have two equal angles: angle FBE = angle FCB and angle FEB = angle FBC. This is correct because both pairs of angles subtend the same arcs.Then, using the similarity ratio, I set up the proportion FB / FC = FE / FB, which led me to FC = 4.Subtracting FE from FC gives CE = 3.Yes, that seems correct. So, the length of CE is 3.

💡Okay, so I have this problem with two sets, A and B. Let me try to understand what each set is.Set A is defined as {X | 1/32 ≤ 2^x ≤ 4}. Hmm, okay, so this is a set of real numbers x such that 2 raised to the power of x is between 1/32 and 4. I think I can rewrite this inequality to find the range of x.Since 2^x is an exponential function, it's increasing, so if 1/32 ≤ 2^x ≤ 4, then taking logarithms base 2 should help. Let me do that.Taking log base 2 of all parts:log₂(1/32) ≤ x ≤ log₂(4)I know that log₂(1/32) is the same as log₂(2^(-5)) because 32 is 2^5, so 1/32 is 2^(-5). Therefore, log₂(2^(-5)) is just -5.Similarly, log₂(4) is log₂(2^2) which is 2.So, set A is all real numbers x such that -5 ≤ x ≤ 2. So, A = [-5, 2].Wait, hold on, the original problem says x ∈ ℤ for part (1). So, for part (1), x is an integer. So, set A when x is integer would be the integers from -5 to 2 inclusive.So, A = {-5, -4, -3, -2, -1, 0, 1, 2}. Let me count how many elements that is. From -5 to 2, that's 8 elements.Now, part (1) asks for the number of non-empty proper subsets of A. Hmm, okay.I remember that the number of subsets of a set with n elements is 2^n. So, for A with 8 elements, the total number of subsets is 2^8 = 256.But we need non-empty proper subsets. Proper subsets are subsets that are not equal to the set itself, and non-empty means they have at least one element.So, the total number of subsets is 256. Subtracting the empty set and the set itself, we get 256 - 2 = 254.So, the number of non-empty proper subsets of A is 254.Wait, let me just make sure I didn't make a mistake. So, A has 8 elements, so 2^8 is 256 subsets. Subtracting the two trivial subsets (empty set and A itself) gives 254. Yeah, that seems right.Okay, moving on to part (2). It says: Find the range of real number values for m if A ⊇ B.Set B is defined as {x | x² - 3m x + 2m² - m - 1 < 0}. Hmm, okay, so this is a quadratic inequality in x. Let me try to factor it or find its roots.The quadratic is x² - 3m x + (2m² - m - 1). Let me see if I can factor this.Looking for two numbers that multiply to (2m² - m - 1) and add up to -3m.Wait, maybe I can factor it as (x - a)(x - b) where a and b are expressions in m.Let me try to factor the quadratic:x² - 3m x + (2m² - m - 1)Let me see if I can factor it as (x - (2m + 1))(x - (m - 1)). Let me check:(x - (2m + 1))(x - (m - 1)) = x² - [(2m + 1) + (m - 1)]x + (2m + 1)(m - 1)Simplify the coefficients:The middle term coefficient is -(2m + 1 + m - 1) = -(3m)The constant term is (2m + 1)(m - 1) = 2m(m) + 2m(-1) + 1(m) + 1(-1) = 2m² - 2m + m - 1 = 2m² - m - 1Yes, that works. So, the quadratic factors as (x - (2m + 1))(x - (m - 1)) < 0.So, set B is the set of x such that (x - (2m + 1))(x - (m - 1)) < 0.Now, to solve this inequality, we need to find the intervals where the product is negative. That happens when one factor is positive and the other is negative.First, let's find the roots of the quadratic, which are x = 2m + 1 and x = m - 1.So, the critical points are x = m - 1 and x = 2m + 1.Now, depending on the values of m, these roots can be in different orders.Case 1: If m - 1 < 2m + 1, which simplifies to m - 1 < 2m + 1 => -1 - 1 < 2m - m => -2 < m.So, if m > -2, then m - 1 < 2m + 1.Case 2: If m - 1 > 2m + 1, which is when m < -2.So, depending on whether m is greater than or less than -2, the order of the roots changes.Also, if m = -2, then m - 1 = -3 and 2m + 1 = -3, so both roots are equal. So, the quadratic becomes a perfect square, and the inequality is (x + 3)^2 < 0, which has no solution because a square is always non-negative. So, set B is empty when m = -2.Okay, so let's consider different cases.Case 1: m > -2Then, m - 1 < 2m + 1.So, the quadratic is negative between the roots. So, B = (m - 1, 2m + 1).Case 2: m < -2Then, m - 1 > 2m + 1.So, the quadratic is negative between 2m + 1 and m - 1. So, B = (2m + 1, m - 1).Case 3: m = -2Then, B is empty set.Now, the problem says A ⊇ B. So, set B must be a subset of A.But A is [-5, 2]. So, depending on whether B is empty or not, we have different conditions.If B is empty, which happens when m = -2, then A ⊇ B is automatically true because the empty set is a subset of any set.If B is not empty, then B must be a subset of A. So, the interval B must lie entirely within A.So, let's handle the cases.Case 1: m > -2Then, B = (m - 1, 2m + 1). We need this interval to be a subset of A = [-5, 2].So, the lower bound m - 1 must be ≥ -5, and the upper bound 2m + 1 must be ≤ 2.So, we have:m - 1 ≥ -5 => m ≥ -4and2m + 1 ≤ 2 => 2m ≤ 1 => m ≤ 0.5But in this case, m > -2, so combining with m ≥ -4, we have m ∈ (-2, 0.5]But wait, we also need to ensure that the interval (m - 1, 2m + 1) is valid, meaning that m - 1 < 2m + 1, which is already satisfied because m > -2.So, in this case, m ∈ (-2, 0.5]Case 2: m < -2Then, B = (2m + 1, m - 1). We need this interval to be a subset of A = [-5, 2].So, the lower bound 2m + 1 must be ≥ -5, and the upper bound m - 1 must be ≤ 2.So, we have:2m + 1 ≥ -5 => 2m ≥ -6 => m ≥ -3andm - 1 ≤ 2 => m ≤ 3But in this case, m < -2, so combining with m ≥ -3, we have m ∈ [-3, -2)Also, we need to ensure that the interval (2m + 1, m - 1) is valid, meaning that 2m + 1 < m - 1, which simplifies to m < -2, which is already satisfied.So, in this case, m ∈ [-3, -2)Case 3: m = -2As before, B is empty set, so A ⊇ B is true.Now, combining all cases:From Case 1: m ∈ (-2, 0.5]From Case 2: m ∈ [-3, -2)From Case 3: m = -2So, overall, m ∈ [-3, 0.5]Wait, let me check if m = -3 is included.When m = -3, B = (2*(-3) + 1, -3 - 1) = (-6 + 1, -4) = (-5, -4). So, B = (-5, -4). Is this a subset of A = [-5, 2]? Yes, because (-5, -4) is within [-5, 2].Similarly, when m = 0.5, B = (0.5 - 1, 2*0.5 + 1) = (-0.5, 2). So, B = (-0.5, 2). Is this a subset of A = [-5, 2]? Yes, because (-0.5, 2) is within [-5, 2].Wait, but when m = 0.5, 2m + 1 = 2*0.5 + 1 = 2, so the upper bound is 2, which is included in A.Similarly, when m = -3, 2m + 1 = -5, so the lower bound is -5, which is included in A.So, m can be from -3 to 0.5, including both endpoints.But wait, when m = -3, B = (-5, -4), which is a valid interval, and when m = 0.5, B = (-0.5, 2), which is also valid.But wait, earlier in Case 1, I had m ∈ (-2, 0.5], and in Case 2, m ∈ [-3, -2). So, combining these, m ∈ [-3, 0.5].But let me double-check if m = -3 is allowed.At m = -3, B = (2*(-3) + 1, -3 - 1) = (-5, -4). So, B is (-5, -4). Since A is [-5, 2], B is a subset of A.Similarly, at m = 0.5, B is (-0.5, 2), which is also a subset of A.So, yes, m can be from -3 to 0.5, including both endpoints.But wait, in Case 1, m > -2, and in Case 2, m < -2, but m = -2 is a separate case where B is empty.So, combining all, m ∈ [-3, 0.5].But let me also check m = -2. At m = -2, B is empty set, which is a subset of A.So, m = -2 is also allowed.Wait, but in my earlier analysis, when m ∈ [-3, -2), and m ∈ (-2, 0.5], and m = -2, so overall m ∈ [-3, 0.5].But wait, when m = -2, it's allowed, so m ∈ [-3, 0.5].Wait, but in Case 2, m ∈ [-3, -2), and in Case 1, m ∈ (-2, 0.5], and m = -2 is allowed, so overall m ∈ [-3, 0.5].Yes, that seems correct.But let me also check m = -3 and m = 0.5.At m = -3, B = (-5, -4), which is within A = [-5, 2].At m = 0.5, B = (-0.5, 2), which is within A.So, yes, m can be from -3 to 0.5, including both endpoints.Therefore, the range of m is [-3, 0.5].Wait, but earlier I thought m ∈ [-3, 0.5], but in the initial analysis, I had m ∈ [-3, -2) and m ∈ (-2, 0.5], and m = -2.So, combining these, m ∈ [-3, 0.5].Yes, that seems correct.Wait, but in the problem statement, part (2) says "Find the range of real number values for m if A ⊇ B."So, the answer is m ∈ [-3, 0.5].But let me just make sure I didn't miss anything.Wait, when m = -3, B = (-5, -4), which is within A.When m = 0.5, B = (-0.5, 2), which is within A.When m = -2, B is empty set, which is a subset of A.When m is between -3 and -2, say m = -2.5, then B = (2*(-2.5) + 1, -2.5 - 1) = (-5 + 1, -3.5) = (-4, -3.5). Wait, that's not correct.Wait, 2m + 1 when m = -2.5 is 2*(-2.5) + 1 = -5 + 1 = -4.And m - 1 = -2.5 - 1 = -3.5.So, B = (-4, -3.5). Is that within A = [-5, 2]? Yes, because -4 > -5 and -3.5 < 2.Wait, but earlier I thought when m < -2, B = (2m + 1, m - 1). So, for m = -2.5, B = (-4, -3.5), which is within A.Similarly, for m = -3, B = (-5, -4), which is within A.For m approaching -2 from below, say m = -2.1, then B = (2*(-2.1) + 1, -2.1 - 1) = (-4.2 + 1, -3.1) = (-3.2, -3.1). Wait, that can't be right because 2m + 1 when m = -2.1 is 2*(-2.1) + 1 = -4.2 + 1 = -3.2, and m - 1 = -2.1 - 1 = -3.1. So, B = (-3.2, -3.1). Wait, that's a very small interval, but it's still within A.Wait, but when m approaches -2 from below, the interval B approaches (-3, -3), which is empty. So, as m approaches -2 from below, B becomes smaller and smaller until it's empty at m = -2.So, in any case, B is a subset of A for m ∈ [-3, 0.5].Wait, but earlier I thought m ∈ [-3, 0.5], but in the initial analysis, I had m ∈ [-3, -2) and m ∈ (-2, 0.5], and m = -2.So, combining these, m ∈ [-3, 0.5].Yes, that seems correct.Therefore, the range of m is from -3 to 0.5, including both endpoints.So, m ∈ [-3, 0.5].But let me just make sure I didn't make a mistake in the quadratic factoring.The quadratic was x² - 3m x + 2m² - m - 1.I factored it as (x - (2m + 1))(x - (m - 1)).Let me expand that to check:(x - (2m + 1))(x - (m - 1)) = x² - (2m + 1 + m - 1)x + (2m + 1)(m - 1)Simplify:= x² - (3m)x + (2m² - 2m + m - 1)= x² - 3m x + (2m² - m - 1)Yes, that's correct.So, the factoring is correct.Therefore, my analysis seems correct.So, the range of m is from -3 to 0.5, including both endpoints.So, m ∈ [-3, 0.5].But wait, in the problem statement, part (2) says "Find the range of real number values for m if A ⊇ B."So, the answer is m ∈ [-3, 0.5].But let me just check if m = -3 is allowed.At m = -3, B = (2*(-3) + 1, -3 - 1) = (-6 + 1, -4) = (-5, -4). So, B = (-5, -4), which is within A = [-5, 2]. So, yes, m = -3 is allowed.Similarly, at m = 0.5, B = (2*0.5 + 1, 0.5 - 1) = (2, -0.5). Wait, no, that can't be right because when m = 0.5, since m > -2, B = (m - 1, 2m + 1) = (0.5 - 1, 2*0.5 + 1) = (-0.5, 2). So, B = (-0.5, 2), which is within A = [-5, 2]. So, yes, m = 0.5 is allowed.Wait, I think I made a mistake earlier when I thought B = (2m + 1, m - 1) when m < -2, but actually, when m < -2, since m - 1 > 2m + 1, the interval is (2m + 1, m - 1). So, for m = -3, it's (-5, -4), which is correct.So, yes, m ∈ [-3, 0.5].Therefore, the range of m is from -3 to 0.5, including both endpoints.So, the final answer for part (2) is m ∈ [-3, 0.5].But wait, in the initial analysis, I had m ∈ [-3, 0.5], but in the problem statement, part (2) says "Find the range of real number values for m if A ⊇ B."So, I think that's correct.Therefore, the range of m is from -3 to 0.5, including both endpoints.

💡Okay, so I have this problem here: "The product of two positive consecutive integers is 812. What is their sum?" Hmm, let's see. I need to find two numbers that are right next to each other, like 5 and 6 or 10 and 11, and when I multiply them together, I get 812. Then, once I find those two numbers, I need to add them up to get the sum.Alright, let's start by understanding what consecutive integers are. Consecutive integers are numbers that follow each other in order, without any gaps. So, if one integer is, say, n, the next one would be n+1. That makes sense. So, in this case, the two numbers we're looking for are n and n+1.The problem says their product is 812. So, mathematically, that would be:[ n times (n + 1) = 812 ]Let me write that out:[ n(n + 1) = 812 ]Expanding that, it becomes:[ n^2 + n = 812 ]Hmm, okay, so this is a quadratic equation. To solve for n, I need to rearrange it into the standard quadratic form, which is:[ ax^2 + bx + c = 0 ]So, subtracting 812 from both sides to get everything on one side:[ n^2 + n - 812 = 0 ]Now, I have the quadratic equation:[ n^2 + n - 812 = 0 ]I need to solve for n. There are a few ways to solve quadratic equations: factoring, completing the square, or using the quadratic formula. Factoring is usually the quickest if the numbers work out nicely, but I'm not sure if that's the case here. Let me see if I can factor this.First, I need two numbers that multiply to -812 and add up to 1 (since the coefficient of n is 1). Hmm, 812 is a bit large, so factoring might be tricky. Let me try to find factors of 812.Breaking down 812:- 812 divided by 2 is 406- 406 divided by 2 is 203- 203 divided by 7 is 29- 29 is a prime numberSo, the prime factorization of 812 is:[ 812 = 2 times 2 times 7 times 29 ]Now, let's list all the factor pairs of 812:- 1 and 812- 2 and 406- 4 and 203- 7 and 116- 14 and 58- 28 and 29Looking at these factor pairs, I need two numbers that multiply to 812 and have a difference of 1 because they are consecutive integers. Let's see:- 28 and 29: 28 times 29 is 812, and they are consecutive integers.Wait, that's exactly what we need! So, n and n+1 are 28 and 29.Therefore, the two consecutive integers are 28 and 29. Now, to find their sum:[ 28 + 29 = 57 ]So, the sum of the two consecutive integers is 57.But just to make sure I didn't make a mistake, let me verify:- 28 times 29 is indeed 812.- 28 plus 29 is 57.Yep, that checks out. So, I think I've got the right answer here.Alternatively, if I wasn't able to factor it easily, I could have used the quadratic formula. The quadratic formula is:[ n = frac{-b pm sqrt{b^2 - 4ac}}{2a} ]For the equation ( n^2 + n - 812 = 0 ), a = 1, b = 1, c = -812.Plugging those values in:[ n = frac{-1 pm sqrt{1^2 - 4(1)(-812)}}{2(1)} ][ n = frac{-1 pm sqrt{1 + 3248}}{2} ][ n = frac{-1 pm sqrt{3249}}{2} ][ n = frac{-1 pm 57}{2} ]So, the two solutions are:[ n = frac{-1 + 57}{2} = frac{56}{2} = 28 ][ n = frac{-1 - 57}{2} = frac{-58}{2} = -29 ]Since we're looking for positive integers, we discard the negative solution. So, n = 28, and the next integer is 29. Their sum is 57.Either way, whether by factoring or using the quadratic formula, I arrive at the same answer. So, I'm confident that the sum of the two consecutive integers is 57.