Use the following definition of absolute value to prove the given statements: If x is a real number, then the absolute value of x , | x | , is defined by: | x | = { x if x ≥ 0 − x if x < 0 For any real number x , | x | ≥ 0 . Moreover, | x | = 0 implies x = 0 . For any two real numbers x and y , | x | ⋅ | y | = | x y | . For any two real numbers x and y , | x + y | ≤ | x | + | y | .
Answers
Answer:
Proved all parts below.
Step-by-step explanation:
As given ,
|x| = \(\left \{ {{x , x\geq 0} \atop {-x, x< 0}} \right.\)
To prove- a) For any real number x , | x | ≥ 0 . Moreover, | x | = 0 ⇒ x = 0
b) For any two real numbers x and y , | x | ⋅ | y | = | x y | .
c) For any two real numbers x and y , | x + y | ≤ | x | + | y | .
Proof -
a)
As given x is a real number
Also , by definition of absolute value of x , we get
| x | ≥ 0
Now,
if |x| = 0
⇒ x = 0 and -x = 0
⇒ x = 0 and x = 0
⇒ x = 0
∴ we get
| x | = 0 ⇒ x = 0
Hence proved.
b)
To prove - | x | ⋅ | y | = | x y |
As we have,
|x| = \(\left \{ {{x , x\geq 0} \atop {-x, x< 0}} \right.\)
|y| = \(\left \{ {{y , y\geq 0} \atop {-y, y< 0}} \right.\)
|xy| = \(\left \{ {{xy , x,y > 0 and x,y < 0} \atop {-xy, x > 0, y< 0 and x <0 , y > 0}} \right.\)
We have 4 cases : i) when x > 0 , y > 0
ii) when x > 0 , y < 0
iii) when x < 0, y > 0
iv) when x < 0, y < 0
For Case I - when x > 0 , y > 0
⇒ |x| = x, |y| = y
⇒|x|.|y| = xy
For Case Ii - when x > 0 , y < 0
⇒ |x| = x, |y| = -y
⇒|x|.|y| = -xy
For Case Iii - when x < 0 , y > 0
⇒ |x| = -x, |y| = y
⇒|x|.|y| = -xy
For Case IV - when x < 0 , y < 0
⇒ |x| = -x, |y| = -y
⇒|x|.|y| = (-x)(-y) = xy
∴ we get , from all 4 cases
| x | ⋅ | y | = | x y |
Hence Proved.
c)
To prove - | x + y | ≤ | x | + | y |
Let
|x| = |x + y - y|
≥ |x + y| - |y| ( Triangle inequality)
⇒ |x| + |y| ≥ |x + y|
Hence Proved.
Related Questions
In the diagram shown below, RQ bisects PRS. The measures of the two congruent angles are (x + 40) and (3x - 20). Solve for "x". P Р (x + 40) 0 (3x - 20° R S VI
Answers
Given:
Angle PRQ = (x + 40)
Angle QRS = (3x - 20)
Here, the line RQ bisects PRS. A bisector can be said to divide an angle into two equal parts.
Thus, angle PRQ = angle QRS
x + 40 = 3x - 20
Let's solve for x.
Subtract 3x from both sides:
x - 3x + 40 = 3x - 3x - 20
-2x + 40 = -20
Subtract 40 from both sides:
-2x + 40 - 40 = -20 - 40
-2x = -60
Divide both sides by -2:
\(\begin{gathered} \frac{-2x}{-2}=\frac{-60}{-2} \\ \\ x\text{ = }30 \end{gathered}\)The value of x is 30
ANSWER:
x = 30
A random sample of 64 students at a university showed an average age of 20 years and a sample standard deviation of 4 years. The 90% confidence interval for the true average age of all students in the university is
Answers
Answer:
The 90% confidence level is \(19.15< L < 20.85\)
Step-by-step explanation:
From the question we are told that
The sample size is \(n = 64\)
The mean age is \(\= x = 20 \ years\)
The standard deviation is \(\sigma = 4 \ years\)
Generally the degree of freedom for this data set is mathematically represented as
\(df = n - 1\)
substituting values
\(df = 64 - 1\)
\(df = 63\)
Given that the level of confidence is 90% the significance level is mathematically evaluated as
\(\alpha = 100 - 90\)
\(\alpha =\)10 %
\(\alpha = 0.10\)
Now \(\frac{\alpha }{2} = \frac{0.10}{2} = 0.05\)
Since we are considering a on tail experiment
The critical value for half of this significance level at the calculated degree of freedom is obtained from the critical value table as
\(t_{df, \frac{ \alpha}{2} } = t_{63, 0.05 } = 1.669\)
The margin for error is mathematically represented as
\(MOE = t_{df , \frac{\alpha }{2} } * \frac{\sigma}{\sqrt{n} }\)
substituting values
\(MOE = 1.699 * \frac{4 }{\sqrt{64} }\)
\(MOE = 0.85\)
he 90% confidence interval for the true average age of all students in the university is evaluated as follows
\(\= x - MOE < L < \= x + E\)
substituting values
\(20 - 0. 85 < L < 20 + 0.85\)
\(19.15< L < 20.85\)
100 Points! Geometry question. Photo attached. Find the measure. Please show as much work as possible. Thank you!
Answers
Answer:
The answer would lie within 31 degrees of MP and also as in PM.
Answer:
central m arc MP=118°
Step-by-step explanation:
here
central m arc MN=2* inscribed m arc MN=2*31=62°
again
central m arc MN+ central m arc MP=180° being linear pair
substituting value
62°+central m arc MP=180°
central m arc MP=180°-62°
central m arc MP=118°
PLEASE HELP MEEEE ILL MARK AS BRAINLIEST
For which table would 70 be the missing value, x?
Answers
Answer:
Table 3 → 5 (5× 14=70)
pls pls help i’ll mark brainliest quickly!!!
Part A: The number of cubic feet of water, y, in a pool 2 hours after the water starts flowing into the pool is a linear function. The points (40,3000) and (60,4000) lie on the line of the function. Calculate the slope of the line.
Part B: What does the slope of the line represent in Part A? Explain in your own words and use compete sentence.
Part C: Explain how the slope formula is related to the definition of slope. Use complete sentences.
Answers
Answer:
m=50
General Formulas and Concepts:
Pre-Alg
Order of Operations: BPEMDAS
Algebra I
Slope Formula:
Step-by-step explanation:
Step 1: Define
Point (40, 3000)
Point (60, 4000)
Step 2: Find slope m
NEED HELP ASAP
6 th grade mathematics question
Answers
Answer:
B, 8 (2a+3b).
Step-by-step explanation:
First, let's let A=1 and B=2. 16a+24b would equal 64, and so would B, which is 8 (2a+3b). Then lastly you solve, and you get 64 for both of them, which means B is equal to that expression.
Answer:
B
Step-by-step explanation:
8(2a) = 16a
8(3b) = 24b
16a + 24b
What is the probability of rolling a two on the first die and a three on the second die?
Answers
Answer:
a dice has 1 two and 1 three
1/6 *1/6 =1/36
For the question of total area of the cuboid is 200cm^.
I understand where we divide 150 by 4.
But why do I need to multiply by 5, when there are 6 faces.
Answers
You need to multiply by 5 instead of 6 because each pair of opposite faces on a cuboid has the same area, so by considering one face from each pair, you ensure that you don't count any face twice.
When calculating the total surface area of a cuboid, you need to understand the concept of face pairs.
A cuboid has six faces, but each face has a pair that is identical in size and shape.
Let's break down the reasoning behind multiplying by 5 instead of 6 in the given scenario.
To find the surface area of a cuboid, you can add up the areas of all its faces.
However, each pair of opposite faces has the same area, so you avoid double-counting by only considering one face from each pair. In this case, you have five pairs of faces:
(1) top and bottom, (2) front and back, (3) left and right, (4) left and back, and (5) right and front.
By multiplying the average area of a pair of faces by 5, you account for all the distinct face pairs.
Essentially, you are considering one face from each pair and then summing their areas.
Since all the pairs have the same area, multiplying the average area by 5 gives you the total surface area.
When dividing 150 by 4 (to find the average area of a pair of faces), you are essentially finding the area of a single face.
Then, by multiplying this average area by 5, you ensure that you account for all five pairs of faces, providing the total surface area of the cuboid.
Thus, multiplying by 5 is necessary to correctly calculate the total surface area of the cuboid by accounting for the face pairs while avoiding double-counting.
For similar question on cuboid.
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I'll add a photo of the problem!
Answers
Answer:
its d
Step-by-step explanation:
the number 30 is not listed in the problem there that is info that is not there
6. Rachel's garden is square in shape. The length of one side of her garden is 52 feet. What is the area of her garden in square feet? Express your answer using exponents.
Answers
Answer:
52^ 2 because this is equal to the area which is 2704 sq ft. So 52 to the second power is your answer.
Step-by-step explanation:
HELP ILL GIVE BRAINLESS DUE SOON
Answers
Answer:
f x 3=8
Step-by-step explanation:
d. 10 + 3у – 2 = 4у — у + 8
Answers
Answer:
3y+8=3y+8
Step-by-step explanation:
10+3y-2=4y-y+8
10+3y-2 combine 10 and -2 and you get 3y+8
4y-y+8 combine the 4y with the -y and you get 3y+8
3y+8=3y+8
help me with this problem please
Answers
Neither of the equation represent the a line that passes through (0, 5) and (4, 15)
How to represent the equation of a line?The equation of a line can be represented in slope intercept form and point slope form as follows:
slope intercept form:
y = mx + b
where
m = slopeb = y-interceptTherefore,
m = 15 - 5 / 4 - 0
m = 10 / 4
m = 5 / 2
let's find y-intercept using (0, 5)
y = 5 / 2 x + b
5 = 5 / 2(0) + b
b = 5
Hence,
y = 5 / 2 x + 5
Let's find the equation in point slope form:
(0, 5)
y - y₁ = m(x - x₁)
y - 15 = 5 / 2(x - 4)
Therefore, neither of the equation represent the line.
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1. UV = 8 and WX = 5
TU=
WU=
TX=
TV=
Answers
All sides of a rhombus have equal measures, so TU = 8. Since a rhombus is a parallelogram, and the diagonals of a parallelogram bisect each other, WU = 10. The diagonals of a rhombus are also perpendicular, meaning they form right angles. Using the Pythagorean theorem, you can find the length of TX. (TX)^2 + (WX)^2 = (WT)^2. Substituting in known values, (TX)^2 + 25 = 64. Solving gives you TX = the square root of 39. TV is double the length of TX, so TV = 2 times the square root of 39.
Graph the following system of equations and find the x-coordinate of the solution
4x + 4y = 44
y = 2x + 2
Answers
Answer:
In order to find the x-coordinate from both equations, we can isolate the y variable on the left-hand side of both equations, and then set them equal to one another, and solve for x (second equation is already done for us):
4x + 4y = 44
4y = 44 - 4x
y = -x + 11
2x + 2 = -x + 11
3x = 9
x = 3
Step-by-step explanation:
Please support my answer.
University Bank pays 5% interest compounded quarterly on regular savings accounts and Rosemont
Savings Bank
pays 5.5% compounded semiannually. Vasily and Oxana Cherchenko had $4,000 to invest
for 4 years. Based on the interest to be earned, which bank offers the better investment?
Answers
well, the interest is going to be the increase factor on both cases, so we can simply check how much each will accumulate to in those 4 years
\(~~~~~~ \stackrel{\textit{\LARGE University Bank}}{\textit{Compound Interest Earned Amount}} \\\\ A=P\left(1+\frac{r}{n}\right)^{nt} \quad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{original amount deposited}\dotfill &\$4000\\ r=rate\to 5\%\to \frac{5}{100}\dotfill &0.05\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{quarterly, thus four} \end{array}\dotfill &4\\ t=years\dotfill &4 \end{cases}\)
\(A = 4000\left(1+\frac{0.05}{4}\right)^{4\cdot 4} \implies A=4000(1.0125)^{16}\implies \boxed{A \approx 4879.56} \\\\[-0.35em] ~\dotfill\)
\(~~~~~~ \stackrel{\textit{\LARGE Rosemont Savings Bank}}{\textit{Compound Interest Earned Amount}} \\\\ A=P\left(1+\frac{r}{n}\right)^{nt} \quad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{original amount deposited}\dotfill &\$4000\\ r=rate\to 5.5\%\to \frac{5.5}{100}\dotfill &0.055\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{semiannually, thus two} \end{array}\dotfill &2\\ t=years\dotfill &4 \end{cases}\)
\(A = 4000\left(1+\frac{0.055}{2}\right)^{2\cdot 4} \implies A=4000(1.0275)^8\implies \boxed{A \approx 4969.52} ~~ \textit{\LARGE \checkmark}\)
The counting number just before C5F sixteen is
Answers
the counting number just before C5F16 is 3,072.
Find the perimeter of the equilateral triangle.
9x16
8-3x
The perimeter is
Answers
The perimeter of the equilateral triangle will be 27x - 48.
How to calculate the perimeter?It should be noted that the perimeter of an equilateral triangle is the addition of all its sides or multiplying by three since they're equal.
Therefore, the perimeter of the equilateral triangle will be:
= 3(9x - 16)
= 27x - 48
Therefore, the perimeter of the equilateral triangle will be 27x - 48.
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Find the perimeter of the equilateral triangle.
9x - 16
Continues (- infinity , infinity) find c
[ cx^2 + 8x
if x < 5
X^3 - cx if x> 5
Answers
It sounds like you're asked to find c such that f(x), defined by
\(f(x)=\begin{cases}cx^2+8x&\text{for }x<5\\x^3-cx&\text{for }x>5\end{cases}\)
is continuous at x = 5.
With the strict inequalities given in the definition, this is not possible. So you probably meant to use ≤ or ≥ in one of the pieces of the definition.
In order for f(x) to be continuous at x = 5, the limit from either side as x approaches 5 must be the same.
We have
\(\displaystyle\lim_{x\to5^-}f(x)=\lim_{x\to5}(cx^2+8x)=25c+40\)
and
\(\displaystyle\lim_{x\to5^+}f(x)=\lim_{x\to5}(x^3-cx)=125-5c\)
Then
\(25c+40=125-5c\implies30c=85\implies c=\dfrac{85}{30}=\boxed{\dfrac{17}6}\)
Solve all this
1) 2x-8=11x-10
2)6x-6=3x+1
3)1/2x+4=1/3x+1
4)-2/5x+1/3=1/2x+1/4
5)5/3x2=1/2x-2/3
Answers
Answer: hii :)
1. 2x-8=11x-10 = 2/9
2. 6x-6=3x+1 = 7/3
3. 1/2x+4=1/3x+1 = -18
4. -2/5x+1/3=1/2x+1/4 = 5/54
5. I dont know the answer to <3
Step-by-step explanation:
Hopefully this helps you,
Sorry for being late on the question.
- Matthew
What strategies do you use to make a ratio table?
Answers
Please help will mark brainiest
Answers
The Perimeter of a triangle is 90 cm. The triangle has side lengths of 2x + 5, 4x - 10, and x + 4. Find the value of x and the length of each side.
Answers
Answer:
This makes no sense as you dont say what type of triangle it is
Step-by-step explanation:
If its a equallateral its x = 89/7
1. Each person in a random sample of 1.026 adults in the United States was asked the following question "Based on what you know about the Social Security System today, what would you like Congress and the President to do during this next year? The response choice and the percentages selecting them are shown below
Completely overhaul the system 19%
Make some major changes 39%
Make some minor adjustments 30%
Leave the system the way it is now
No opinion 1%
Find a 95% confidence interval for the proportion of all United States adults who would respond "Make some major changes to the question
a. Identify the variables needed to solve the problem.
b. Can a normal distribution be used to approximate this data Justify your evidence. c. Find the standard deviation
d. Calculate the point estimate and margin of error
e. calculate the confidence interval
Answers
Answer:
Step-by-step explanation:
a. Variables needed to solve the problem:
Sample size: n = 1,026
Proportion of the sample that responded "Make some major changes": p = 0.39
Confidence level: 95%
b. To determine if a normal distribution can be used to approximate the data, we need to check if the sample size is large enough to meet the requirements for a normal approximation. The sample size should be at least 10 times larger than the number of successes (np) and 10 times larger than the number of failures (n(1-p)). In this case, we have:
np = 1026 x 0.39 = 399.14
n(1-p) = 1026 x 0.61 = 626.86
Both np and n(1-p) are greater than 10, so we can assume that a normal distribution can be used to approximate the data.
c. The standard deviation of the proportion can be calculated using the following formula:
standard deviation = sqrt(p(1-p) / n)
standard deviation = sqrt(0.39 x 0.61 / 1026) = 0.024
d. The point estimate of the proportion of all United States adults who would respond "Make some major changes" is simply the sample proportion, which is p = 0.39. The margin of error can be calculated using the following formula:
margin of error = z* * standard deviation
where z* is the z-score associated with the 95% confidence level. Using a standard normal distribution table or a calculator, we find that the z-score for a 95% confidence level is approximately 1.96. Therefore:
margin of error = 1.96 * 0.024 = 0.047
e. The confidence interval can be calculated using the following formula:
confidence interval = point estimate ± margin of error
confidence interval = 0.39 ± 0.047
confidence interval = (0.343, 0.437)
Therefore, we are 95% confident that the proportion of all United States adults who would respond "Make some major changes" is between 0.343 and 0.437.
¿De qué número 64 es el 80%?
Answers
Order the values from least (on top) to greatest (on bottom).
64%
0.65
5/8
Answers
Convert them all into decimals. Percent is just 0.--, and the fraction you can calculate out which gives: 0.625=5/8,0.64=64%
5/8
64%
0.65
Help pls number 1.......
Answers
Answer:
the answer is 0.3
Step-by-step explanation:
this is because 7 2/5 is equal to 2.8 then if you add all of the number together you will get 24.7 meaning that if you add 0.3 to that you will have your answer of 25
don't have to answer all just at least 1 pls and write yes if the statement is a proportion or no if it is not a proportion.
2:3 = 9:4
3/8 = 12/32
10/12 = 20/25
4/9 = 17/38
Answers
Answer:
3/8 = 12/32 is a proportion
Step-by-step explanation:
2:3 and 9:4 is not a proportion, 10/12 and 20/25 is not a proportion, 4/9 and 17/38 is not a proportion
what is the slope of the line
Answers
Answer: -3/4
Step-by-step explanation:
(-1,1)
What is 2+2+2-2+4-8?
Answers
Answer:
Answer- 0
Step-by-step explanation:
2+2+2-2+4-8= 4-2+4-8=0
hope this helps :)