What is the example of Triangle Inequality theorem 2?

P/Q is in the lowest form as the when the HCF is 1 and if we divide it by 1 P/Q remains same so it is the lowest form ,if P and Q have no common factor other than 1.

It is possible to express rational numbers in the form p/q, where p and q are integers and q≠0.

If there is just one other common factor between the denominator and numerator of a rational number, that number is said to be in its lowest form. In other words, if the HCF of a and b is 1 and a and b are relatively prime, then we can state that a rational number a/b is considered to be in its simplest form.

as described in the above contion in order to reduce any rational number to its lowest form we need to divide both numerator and denominator by the HCF of p,q . suppose if the HCF of p,q is 1 then divide p and q 1 won't change the value of the rational number.

so, now it is shown that the given rational number can't be further deduced and hence we conclude that a rational number is in lowest form when HCF of its numerator and denominatior is equal to 1.

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Complete question:

Write the reason why a rational number written in the form of p/q is said to be in lowest form if P and Q have no common factor other than 1.

On the list of 98 factors, there are 1, 2, 7, 14, 49, and 98 factors. All of the factors of 98 are 1, 2, 7, 14, 49, and 98.

A factor is a number that completely divides another number.

To put it another way, if adding two whole numbers results in a product, then the numbers we are adding are factors of the product because the product is divisible by them.

The two components that makeup 98 are: (1, 98), (2, 49), and (7, 14). 1, 2, and 7 make up the first three factors of 98.

So, in order to get what we can get into 98, we need to list down the factors of 98.

There are 1, 2, 7, 14, 49, and 98 on the list of 98 factors.

1, 2, 7, 14, 49, and 98 are all factors of 98.

Therefore, on the list of 98 factors, there are 1, 2, 7, 14, 49, and 98 factors. All of the factors of 98 are 1, 2, 7, 14, 49, and 98.

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Answer:$16.15

Step-by-step explanation:

The wholesale price:

$8.50

Selling Price:

$8.50 + 90% = $8.50 *1.9 = $16.15

Pair of shoes sold for $16.15 in the store

HOPE THIS HELPS!

The value of k is ±4 so that the roots of the quadratic equation 9x² + 3kx +4 are equal

A second-degree polynomial equation, which must contain at least one squared component, is what is known as a quadratic. Quadratic equations is another name for them. The quadratic equation's general form is as follows: ax² + bx + c = 0

Quadratic equation,  ax2 + bx + c = 0 has equal roots if (b2 - 4ac) = 0

Hence in quadratic equation, 9x2 + 3kx +4 = 0 has equal roots if

(3k)² - 4×9×4  = 0

Hence 9k² = 144   or  k² = 16   or  k = ±4

In order for the roots of the quadratic equation 9x2 + 3kx + 4 to be equal, the value of k must be 4.

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The sample size of 200 can be represented by the equation n1 + n2 + n3 + n4, where n1, n2, n3, and n4 represent the number of first-year, sophomore, junior, and senior students surveyed, respectively.

The sample of 200 students surveyed by the statistics students was composed of 50 first-year students, 50 sophomores, 50 juniors, and 50 seniors. This could be represented by the following formula:

Sample Size = 50 First-Years + 50 Sophomores + 50 Juniors + 50 Seniors

In order to break down this sample size into a mathematical formula, the students can use the following equation:

Sample Size (SS) = n1 + n2 + n3 + n4

Where n1, n2, n3, and n4 represent the number of first-year, sophomore, junior, and senior students surveyed, respectively.

Therefore, using the original sample size, the equation would be:

SS = 50 First-Years + 50 Sophomores + 50 Juniors + 50 Seniors

SS = n1 + n2 + n3 + n4

SS = 50 + 50 + 50 + 50

SS = 200

This shows that the sample size of 200 can be represented by the equation n1 + n2 + n3 + n4, where n1, n2, n3, and n4 represent the number of first-year, sophomore, junior, and senior students surveyed, respectively.

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Answer:

x = 30.6 ft

A = 58.1°

Step-by-step explanation:

[tex]\boxed{\begin{minipage}{9 cm}\underline{Pythagoras Theorem} \\\\$a^2+b^2=c^2$\\\\where:\\ \phantom{ww}$\bullet$ $a$ and $b$ are the legs of the right triangle. \\ \phantom{ww}$\bullet$ $c$ is the hypotenuse (longest side) of the right triangle.\\\end{minipage}}[/tex]

As the triangle is a right triangle, use Pythagoras Theorem to find length x.

[tex]\implies x^2+19^2=36^2[/tex]

[tex]\implies x^2+361=1296[/tex]

[tex]\implies x^2=935[/tex]

[tex]\implies x=\sqrt{935}[/tex]

[tex]\implies x=30.6\; \sf ft \;\; (3\;s.f.)[/tex]

[tex]\boxed{\begin{minipage}{9 cm}\underline{Cos trigonometric ratio} \\\\$\sf \cos(\theta)=\dfrac{A}{H}$\\\\where:\\ \phantom{ww}$\bullet$ $\theta$ is the angle. \\ \phantom{ww}$\bullet$ $\sf A$ is the side adjacent the angle. \\\phantom{ww}$\bullet$ $\sf H$ is the hypotenuse (the side opposite the right angle). \\\end{minipage}}[/tex]

From inspection of the given right triangle:

Substitute the values into the formula and solve for A:

[tex]\implies \sf \cos A=\dfrac{19.0}{36.0}[/tex]

[tex]\implies \sf A= \cos^{-1}\left(\dfrac{19}{36}\right)[/tex]

[tex]\implies \sf A=58.1^{\circ}\;\;(3\;s.f.)[/tex]

Answer:

2.5n ≤ 15

Step-by-step explanation:

Let n = number of times

Each toll is $2.50, so the toll for n trips is n times $2.50, or 2.5n.

The total amount of money available is $15, so 2.5n must be less than or equal to 15.

Inequality:

2.5n ≤ 15

(In case you're asked next, here is the solution of the inequality:

Divide both sides by 2.5.

2.5n/2.5 ≤ 15/2.5

n ≤ 6

A person can drive up to 6 times with $6.)

The real estate agency made $24,375 on the house. The agent earned $6,093.75 in commission.

The correct solution of the given function is g'(t)=20 for some t in the interval. The correct option is (B).

Definition of function- Function, in mathematics, an expression, rule, or law that defines a relationship between one variable (the independent variable) and another variable (the dependent variable). Functions are ubiquitous in mathematics and are essential for formulating physical relationships in sciences.

Given that g(10)= 220, g(0)=20

Explanation: A function from a set X to a set Y assigns to each element of X exactly one element of Y. The set X is called the domain of the function and the set Y is called the codomain of the function.

For a given function, y = F(x), if the value of y is increasing on increasing the value of x, then the function is known as an increasing function and if the value of y is decreasing on increasing the value of x, then the function is known as a decreasing function.

So putting values in condition of differentiation for increasing function we get:

=[tex]\frac{g(10)-g(0)}{10-0}[/tex]=[tex]\frac{220-20}{10}[/tex]=[tex]\frac{200}{10}[/tex]=20

The correct solution for given function is g'(t) = 20 for some t in the interval.

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The nature of the roots of a given quadratic equation √3x²+10x+7√3=0 is real and distinct because of the value of b²-4ac>0.

where a, b, and c are the coefficients in the standard form of the quadratic equation ax²+bx+c=0

Given quadratic equation is  √3x²+10x+7√3=0   on comparing with the standard form of a quadratic equation ax²+bx+c=0. Hence a=√3,b=10,

and c=7√3

Now  b²-4ac=10²-4×√3×7√3=100-28×3=16>0.

here the value of b²-4ac is greater than 0.

We know if the value of b²-4ac then the roots of the given equations are real and distinct.

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Answer:

See below for proof.

[tex]S_{500}=125250[/tex]

Step-by-step explanation:

Given arithmetic series:

Therefore:

[tex]S_n=1+2+3+...+(n-2)+(n-1)+n[/tex]

[tex]S_n=1+(1+1)+(1+2)+...+(1+n-3)+(1+n-2)+(1+n-1)[/tex]

[tex]S_n=1+(1+1)+(1+2(1))+...+(1+(n-3)(1))+(1+(n-2)(1))+(1+(n-1)(1))[/tex]

Let:

Therefore:

[tex]S_n=a+(a+d)+(a+2d)+...+(a+(n-3)d)+(a+(n-2)d)+(a+(n-1)d)[/tex]

Reverse the order:

[tex]S_n=(a+(n-1)d)+(a+(n-2)d)+(a+(n-3)d)+...+(a+2d)+(a+d)+a[/tex]

Add the two expressions for Sₙ:

[tex]2S_n=(2a+(n-1)d)+(2a+(n-1)d)+(2a+(n-1)d)+...+(2a+(n-1)d)[/tex]

Therefore, the term (2a + (n – 1)d) has been repeated n times:

[tex]2S_n=n(2a+(n-1)d)[/tex]

Divide both sides by 2:

[tex]S_n=\dfrac{1}{2}n(2a+(n-1)d)[/tex]

[tex]S_n=\dfrac{1}{2}n(a+a+(n-1)d)[/tex]

Replace  a with  a₁  (first term) and a + (n – 1)d  with  aₙ (last term):

[tex]S_n=\dfrac{1}{2}n(a_1+a_n)[/tex]

To find the sum of the series 1 + 2 + 3 + ... + 500, substitute the following values into the formula:

Therefore:

[tex]\implies S_{500}=\dfrac{1}{2}(500)(1+500)[/tex]

[tex]\implies S_{500}=250(501)[/tex]

[tex]\implies S_{500}=125250[/tex]

Knowing merely angle-angle-angle (AAA) does not work, because it can result in triangles that are similar but not congruent.

A set of triangles must have equal sides and angles in order for them to be congruent. The sides of the triangles may or may not be equal in the case of a triangle with all of the corresponding angles equal, or the AAA condition. Only if the two triangles are similar would the congruence be true for two triangles with identical respective angles.

So, knowing merely angle-angle-angle (AAA) does not work, because it can result in triangles that are similar but not congruent.

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The answer to the above question will be as follows:

The correct option is option c.) The graph will be a solid line with a y-intercept of seven and a slope of negative four. The graph will be shaded above the line.

We solve the question in the following way,

Given inequality is y≥7x- 4

Let us first consider the linear equality of the equation as

y= 7x- 4

When we consider this with the general slope form of a straight line given by: y=mx+c, we get

m=7 and c= -4

Now, we know that for the ‘greater than or smaller than equal to’ sign, the graph is mapped using a solid line and is shaded away from the origin of the graph (0,0).

Thus for this graph, we will use a solid line to represent the line y=7x- 4 and shade the graph above the line. Therefore option (c.) is correct.

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Note that the full question is:

Which description of the graph of the linear inequality y ≥ 7x – 4 is correct?

a.) The graph will be a dashed line with a y-intercept of negative four and a slope of seven. The graph will be shaded below the line.

b.) The graph will be a solid line with a y-intercept of negative four and a slope of seven. The graph will be shaded above the line.

c.) The graph will be a solid line with a y-intercept of seven and a slope of negative four. The graph will be shaded below the line.

d.) The graph will be a dashed line with a y-intercept of seven and a slope of negative four. The graph will be shaded above the line.

Answer:

(0,2)

Step-by-step explanation:

The equations must be derived from the graph:

First, find the slope of f(x):

[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex] let [tex](x_1,y_1)=(-3,3)[/tex] and [tex](x_2,y_2)=(3,1)[/tex]

[tex]m=\frac{(1)-(3)}{(3)-(-3)}\\m=\frac{-2}{6}\\m=-\frac{1}{3}[/tex]

Then use [tex]y-y_1=m(x-x_1)[/tex]

[tex]y-3=-\frac{1}{3}[x-(-3)]\\y-3=-\frac{1}{3}x-1\\y=-\frac{1}{3}x+2\\f(x)=-\frac{1}{3}x+2[/tex]

For g(x), let:

[tex](x_1,y_1)=(-3,0)\\(x_2,y_2)=(0,2)[/tex]

[tex]m=\frac{(2)-(0)}{(0)-(-3)}\\m=\frac{2}{3}[/tex]

[tex]y-0=\frac{2}{3}[x-(-3)]\\y=\frac{2}{3}x+2\\g(x)=\frac{2}{3}x+2[/tex]

Now, to solve for x, allow the two expressions written in terms of x equal each other.

[tex]-\frac{1}{3}x+2=\frac{2}{3}x+2\\(-\frac{1}{3}x+2)+\frac{1}{3}x=(\frac{2}{3}x+2)+\frac{1}{3}x\\2=x+2\\(2)-2=(x+2)-2\\x=0[/tex]

To solve for y, substitute 0 for x in f(x):

[tex]y=-\frac{1}{3}x+2\\y=-\frac{1}{3}(0)+2\\y=0+2\\y=2[/tex]

So, the solution to this system of equations would be the ordered pair (0,2).  This solution is seen on the graph as the intersection of the two lines.

The word or phrase no which best describes this kind of advertising is inaccurate.

The correct answer option is option B

Mode refers to a statistical data which occurs most in a given set of data.

Advertisement can be defined as a commercial solicitation by a producer to the public in order to buy his product.

Given data:

$7, $8, $8, $15, $15, $16, $17, $18, and $20

Hence, it can be said that the mode of the most popular meals is not $8

Therefore, using $8 as a kind of advertisement for the mode of most popular meals is inaccurate

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As per the value of angle of the triangle, we have identified that the ASA is congruent.

In math the term triangle refers the three sided triangle that contains the three angles, sides and edge.

Here we need to find the process to know if ASA is congruent.

According to the definition of triangle, here we must know the definition of congruent.

The term congruent is defined as if all three corresponding sides are equal and all the three corresponding angles are equal in measure.

In this theorem, we have to know that any two angles and the side included between the angles of one triangle are equivalent to the corresponding two angles and side included between the angles of the second triangle then those two triangles are said to be congruent by ASA rule.

Based on this rule, we have identify that if the two triangle have the same angle, then it said to be congruent as per the ASA rule.

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the equation in slope-intercept form that represents the total number of riders over time is: Y = 33x + 132

To find the equation in slope-intercept form that represents the total number of riders over time, we need to find the slope and the y-intercept of the line. The slope of a line is the change in y divided by the change in x, or the rise over the run. In this case, the change in y between the first and second times is 165 - 132 = 33, and the change in x is 1 - 0 = 1. So the slope of the line is 33/1 = 33. The y-intercept is the point at which the line crosses the y-axis, or the value of y when x is 0. In this case, the y-intercept is 132, since that is the total number of riders at the beginning of Tara's shift when x is 0.

Therefore, the equation in slope-intercept form that represents the total number of riders over time is:

Y = 33x + 132

Note that the slope (33) is the coefficient of the x term, and the y-intercept (132) is the constant term.

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After rotating 90 degrees anticlockwise, B's new coordinates are (y, -x).

By switching the x and y coordinates and negating the new x coordinate, we can rotate a point (x, y) 90 degrees clockwise around the origin.

After rotating the triangle 90 degrees anticlockwise around the origin, note the new coordinates of B.

Let (x, y) be point B's coordinates.

Fllowing a 90° anticlockwise turn:

B'x, the modified x coordinate, will now be y.

B'y, the modified y coordinate, will now be -x.

Therefore, B's new coordinates are (y, -x).

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Answer:

  (c)  two right triangles, each with both legs 40 inches long

Step-by-step explanation:

You want to know which pair of triangles are congruent, given their descriptions.

There are basically 4 theorems that allow you to claim congruence of triangles:

  AAS, ASA, SAS, SSS

A fifth one applies to right triangles only: HL.

These tell you that you need to have three matching parts: 3 sides, 2 sides and the angle between (except right triangle), or 1 side and 2 angles.

An isosceles triangle with only one or two sides defined is not sufficient.

A right triangle with only the hypotenuse defined is not sufficient.

A right triangle with two legs and the (right) angle between them has a sufficient number of matching parts to be able to claim congruence to another triangle similarly defined.

The probability that it came from the first company is  1/2.88⁹.

Probability is the chance of occurrence of a certain event out of the total no. of events that can occur in a given context.

Conditional probability is a term used in probability theory to describe the likelihood that one event will follow another given the occurrence of another event.

Given, A plumbing contractor obtains 60% of her boiler circulators from Company 1 whose defect rate is 0.005 so 40% of her boiler circulators from Company 2 whose defect rate is 0.01.

Now, The probability of the defect rate of company 1 is P(D1) = 0.005/60.

And, The probability of the defect rate of company 1 is P(D1) = 0.01/40.

Therefore, The probability of that it came from the first company is,

= (0.005/60)×( 0.01/40)/60.

= (1/12000)×(1/4000)/50.

= 1/2.88⁹

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Set is a collection of unique objects. The objects are unique, that is there is no repetition in set, making it distinct from a group. A set is well defined.

In mathematics, the objects belonging to a set are called the elements of that particular set.

These elements will be listed in a pair of curly brackets. For example, A is a set of first ten natural numbers, that is its elements are 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10. According to the notation

A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

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The required median of the first ten natural numbers is 5.5.

The median is the number in the middle of a list of numbers in ascending order, from lowest to highest.

The median is the middlemost element if the total number of elements in the list is odd, and the median is the average of the middlemost two values if the total number of elements in the list is even.

The list of the first 10 natural numbers is: 1,2,3,4,5,6,7,8,9,10

Here is the total number of elements in the list is even.

And the two middlemost elements are 5 and 6.

The required median is: 5+6 /2 = 11/2 = 5.5

Hence, the required median of the first ten natural numbers is 5.5.

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Answer:

Step-by-step explanation:

Translations, rotations, and reflections can prove that figures are congruent because these transformations preserve angle measures and distances of the figures. These transformations are also called rigid motions.

The square root of some number 'X' is a number which, when you multiply it by itself, the answer is 'X'.

The square root of a number is the factor that we can multiply by itself to get that number. The symbol for square root is [tex]\sqrt{x}[/tex] . Finding the square root of a number is the opposite of squaring a number.

In mathematics, a square root of a number x is a number y such that y² = x; in other words, a number y whose square (the result of multiplying the number by itself, or y ⋅ y) is x.

For example, 4 and −4 are square roots of 16, because 4² = (−4)² = 16.

Thus , the square root of some number 'X' is a number which, when you multiply it by itself, the answer is 'X'.

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Both Carlos and Irene offered answers equivalent to the original expression.

What is an expression?

Mathematical expressions consist of at least two numbers or variables, at least one arithmetic operation, and a statement. It's possible to multiply, divide, add, or subtract with this mathematical operation. An expression's structure is as follows: Expression: (Math Operator, Number/Variable, Math Operator)

Given expression is

28x − 8.

The prime factors of 28 is 2 and 7.

Rewrite 28 as the product prime factors:

28 = 7 × 2 × 2.

The prime factors of 8 is 2.

Rewrite 8 as the product prime factors:

8 = 2 × 2 × 2.

Replace 28 = 7 × 2 × 2 and 8 = 2 × 2 × 2 in the given expression:

28x − 8

= (7 × 2 × 2) x − (2 × 2 × 2)

Taking common 2

= 2[ (7 × 2)x − (2 × 2)] = 2(14x − 4)

Again taking common 2:

= 2× 2[ 7x − 2]

= 4(7x − 2)

Thus 2(14x − 4) and 4(7x − 2) is the equivalent expression of 28x − 8.

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Answer:

Step-by-step explanation:

Rohan is a generous person who cares deeply for his maid and her two children. On Rohan's birthday, he plans to give both of them the same number of dresses. This situation can be solved using Euclid’s axiom that states “things which are equal to the same thing are also equal to one another”. Therefore, after Rohan's birthday each child will have an equal number of dresses as they were given an identical amount from him.

This kind gesture by Rohan demonstrates his commitment towards taking care of those around him and providing fairness in all situations possible - it serves as a reminder that we should always strive for equality among people regardless of their backgrounds or social statuses in life! By following this axiom, everyone can benefit from having access to resources no matter what their circumstances may be; thus creating a more equitable society overall.

The act done by Rohana is praiseworthy and must serve as an example for others too so they understand how important it is treat everyone equally regardless if they belong to different castes or classes in our society! We should take measures like these on regular basis so that true justice prevails everywhere without any discrimination whatsoever- this would make sure everybody has access basic necessities such us clothes just like Rohana provided his maid's kids with new sets on his own birthday!

The number of golf balls that are in each bucket would be = bucket X = 62, bucket y = 34

Division is one of the major arithmetic operation that is used to know the number of groups that can be placed under data sets when shared.

The number of golf balls that are picked up at driving range = 96

The number of buckets present= 2( X and y)

Bucket X = 28 + a

Bucket y = a

To calculate for each bucket first find a.

But 96 = 28+ a+ a

96 = 28 + 2a

2a = 96- 28

2a = 68

a = 68/2 = 34

Bucket X = 28+34 = 62

Bucket y = 34.

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From the given options ASA is not a test for similarity. Hence option D is the correct answer.

Theorem for similarity of triangles states that we can say two triangles are similar if their corresponding angles are same and the corresponding sides are same. And a result of that their corresponding remaining sides and angles will be also equal.

Hence the possible similarity postulates for two triangles to be similar are AAA(angle angle angle) SAS(side angle side),SSS(side side side)

Therefore the correct option is D. angle side angle (ASA) is the one which does not fall under similarity test.

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The fraction that has the smallest denominator is [tex]\frac{9}{11} < =\frac{5}{6} < =\frac{11}{13}[/tex]

Notice that the fractions seem to have the form  n / [ n + 2]    where n is an integer

Suppose that there exists a fraction such that

9/11 < n / [n + 2]  < 11/13

We can split this into two inequalities

Looking at the inequality on the left....we have

9[n + 2] < 11n    

9n + 18 < 11n

18 < 2n

9 < n    or

n > 9

Looking at the inequality on the right.... we have

13(n) < 11[n + 2]

13n < 11n + 22

2n < 22

n < 11

This implies that

9 < n < 11

So...n = 10 is the only integer that satisfies this

And n + 2 = 12

So....the fraction is 10/12  = 5/6  

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To find the median of a set of numbers, you first need to put the numbers in order from smallest to largest. Once the numbers are ordered, the median is the middle number in the list.

In this case, the ordered list of numbers is: 2, 3, 3, 4, 5, 6, 7.

The median of this list is 4.

Sorting all the data points and selecting the central one are required to determine the middle number (or if there are two middle numbers, taking the mean of those two numbers). Example: The number 4 is in the middle when the numbers are grouped (1, 4, 7), making it the median of the numbers (4, 1 and 7).

By summing up the center values and dividing them by two, you may obtain the mean and, from there, the median.

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