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Stolb23 [73]
3 years ago
14

PLEASE HELP 34 POINTS

Mathematics
2 answers:
Taya2010 [7]3 years ago
7 0

Consider the two expressions, (a + 1)² and (a + 1)³.

No matter what the number is, a² ≥ 0. Even if you square a negative number, you still get a positive number. We include the "or equal to" since 0² = 0. Since a² ≥ 0 is always true, then we can manipulate this inequality.

a² ≥ 0

a² + 2a + 1 ≥ 0 + 2a +1 by adding 2a +1 to both sides

(a + 1)² ≥ 2a + 1 as a² + 2a + 1 factors into (a +1)²

(a + 1)² ≥ 1 If it's more than 2a +1, it has to be more than 1

(a + 1)² ≥ 0 If it's more than one, it's more than zero since 1 > 0.

So we can conclude that (a + 1) ≥ 0 is always true.

Now let's look at (a + 1)³. We know that from before (a + 1)² ≥ 0. A tempting thing to say is that if you multiply by both sides by (a + 1) then both will be more than zero. Doing so isn't right.

What we instead must do is use cases. Either a is positive, negative, or zero.

Case 1: a = 0

When a = 0, (a + 1)³ = 1³ = 1 and 1 > 0. Thus (a + 1)³ ≥ 0,

Case 2: a > 0

When a > 0, a² > 0 from multiplying both sides by a. Do it a second time and a³ > 0. Then, if we add terms to both sides like in the a² example, we have this:

a³ > 0

a³ + 3a² > 3a² by adding 3a² to both sides

a³ + 3a² + 3a > 3a² + 3a by adding 3a to both sides

a³ + 3a² + 3a + 1 > 3a² + 3a + 1 by adding 1 to both sides

(a + 1)³ > 3a² + 3a + 1 by factoring the left side

(a + 1)³ > 1 Since a > 0 by assumption then 3a > 0 and 3a² > 0 and their sum is more than zero too

(a + 1)³ > 0 Since 1 > 0

Case 3: a < 0

Since a < 0 then a² > 0 (minus times minus is plus), but a³ < 0 through a similar multiplication.

Let b = a + 1. If a < 0 then a + 1 < 1 by adding 1 to both sides and b < 1 by back substitution. So b³ < 1³ by cubing both sides and b³ < 1 since 1³ =1.

b³ < 1

(a+1)³ < 1 We chose a + 1 = b.

When a < 0, we can conclude that (a + 1)³ < 1. When a ≥ 0, then (a +1)³ ≥ 0.

However for all a, (a +1)² ≥ 0. Thus, we have our sometimes truth. That when we choose a to be negative we have that the (a + 1)² and (a + 1)³ are of opposite signs.

maks197457 [2]3 years ago
3 0

Answer:

Consider the two expressions, (a + 1)² and (a + 1)³.No matter what the number is, a² ≥ 0. Even if you square a negative number, you still get a positive number. We include the "or equal to" since 0² = 0. Since a² ≥ 0 is always true, then we can manipulate this inequality.a² ≥ 0a² + 2a + 1 ≥ 0 + 2a +1 by adding 2a +1 to both sides(a + 1)² ≥ 2a + 1 as a² + 2a + 1 factors into (a +1)²(a + 1)² ≥ 1 If it's more than 2a +1, it has to be more than 1(a + 1)² ≥ 0 If it's more than one, it's more than zero since 1 > 0.So we can conclude that (a + 1) ≥ 0 is always true.Now let's look at (a + 1)³. We know that from before (a + 1)² ≥ 0. A tempting thing to say is that if you multiply by both sides by (a + 1) then both will be more than zero. Doing so isn't right.What we instead must do is use cases. Either a is positive, negative, or zero.Case 1: a = 0When a = 0, (a + 1)³ = 1³ = 1 and 1 > 0. Thus (a + 1)³ ≥ 0,Case 2: a > 0When a > 0, a² > 0 from multiplying both sides by a. Do it a second time and a³ > 0. Then, if we add terms to both sides like in the a² example, we have this:a³ > 0a³ + 3a² > 3a² by adding 3a² to both sidesa³ + 3a² + 3a > 3a² + 3a by adding 3a to both sidesa³ + 3a² + 3a + 1 > 3a² + 3a + 1 by adding 1 to both sides(a + 1)³ > 3a² + 3a + 1 by factoring the left side(a + 1)³ > 1 Since a > 0 by assumption then 3a > 0 and 3a² > 0 and their sum is more than zero too(a + 1)³ > 0 Since 1 > 0Case 3: a < 0Since a < 0 then a² > 0 (minus times minus is plus), but a³ < 0 through a similar multiplication.Let b = a + 1. If a < 0 then a + 1 < 1 by adding 1 to both sides and b < 1 by back substitution. So b³ < 1³ by cubing both sides and b³ < 1 since 1³ =1.b³ < 1(a+1)³ < 1 We chose a + 1 = b.When a < 0, we can conclude that (a + 1)³ < 1. When a ≥ 0, then (a +1)³ ≥ 0.However for all a, (a +1)² ≥ 0. Thus, we have our sometimes truth. That when we choose a to be negative we have that the (a + 1)² and (a + 1)³ are of opposite signs.

Step-by-step explanation:

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Drupady [299]

Givens

Let the number of students in the class be x

Let the number of pieces of gum she gave out be 3x

Equation

3x + 8 = 168 This will not work out evenly. Let's try x - 1. The reason for that is because she may not give out anything to herself.

3(x - 1) + 8 = 168      This doesn't work either.

Well we have to choose. It's a rounding problem.

3x + 8 = 168     Subtract 8 from both sides.

3x = 168 - 8      Combine

3x = 160           Divide by 3 on both sides.

x = 160 / 3

x = 53.333333333

Since that can't be, we could say there were 53 students.

3x

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3 years ago
Integral from negative 1 to 1 left parenthesis 2 x squared plus 5 right parenthesis dxI. Using the trapezoidal rulea. Estimate t
ANEK [815]

Split the integration interval [-1, 1] into 4 subintervals:

[-1, -1/2], [-1/2, 0], [0, 1/2], [1/2, 1]

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f(x)=2x^2+5\implies\begin{cases}f(-1)=7\\f\left(-\frac12\right)=\frac{11}2\end{cases}

So over this subinterval, the trapezoid contributes an area of

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3 0
2 years ago
4 OT
rewona [7]

Answer:

A. <6

Step-by-step explanation:

Exterior angle of this geometric shape is 6

6 0
3 years ago
Read 2 more answers
A jar contains 4 purple, 3 blue, and 1 gold ticket. Three tickets are selected without replacement.
sveta [45]

Answer:

a) 1/14

b) 1/56

c) 1/28

d) 5/28

Step-by-step explanation:

a) Since tickets are being selected without replacement, this means that the pool of outcomes diminishes with each selection. For the first pick, there are a total of 8 tickets to choose from, and 4 of those are purple. Therefore, the probability that the first ticket is purple is 4/8=1/2. For the second ticket, the total number of remaining tickets is 7, and the number of purple tickets is 3, meaning that the probability is 3/7. For the final ticket, 6 remaining tickets and 2 of them being purple yields a probability of 2/6=1/3. Multiplying all of these together, you get 1/2*3/7*1/3=1/14.

b) Similarly to the previous problem, the first probability is 3/8, followed by 2/7, and then by 1/6. Multiplying these together, you get a probability of 1/56.

c) The probability that the first two tickets are purple are 1/2 and 3/7, as we have already calculated. For the final ticket, there are 6 possible remaining with 1 of them being gold. This means a probability of 1/6, which yields a total of 1/28.

d) Since there are 5 tickets originally that aren't blue, the probability for the first pick is 5/8, followed by 4/7 and 3/6=1/2. Multiplying these together, you get a final answer of 5/8*4/7*1/2=5/28.

Hope this helps!

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3 years ago
Write an addition statement that could be<br> represented by the number line below.
Sholpan [36]

Answer:

-1+5

Step-by-step explanation:

Since you want an addition problem:

it goes down to -1 (represented by the arrow going to the left) and then from -1 it goes up 5 (represented by t he arrow going to the right)

7 0
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