The quadratic function that fits the given points is: f(x) = (4/3)x² - (10/3)x + 1
By using the simplification formula
f(x) = ax² + bx + c
where a, b, and c are constants
a(-1)² + b(-1) + c = 1/3
a(0)² + b(0) + c = 1
a(1)² + b(1) + c = 3
a(2)² + b(2) + c = 9
Simplifying each
a - b + c = 1/3
c = 1
a + b + c = 3
4a + 2b + c = 9
We can solve this using any method like substitution, elimination
a - b + c = 1/3
a + b + c = 3
2a + 2c = 9/3
Adding the first two equations 2a + 2c = 10/3
Subtracting the third equation
b = 5a/3 - 2
a - (5a/3 - 2) + 1 = 1/3
a = 4/3
Finally, we can substitute a = 4/3 and b = 5a/3 - 2, and c = 1 into the standard form of the quadratic equation to get: f(x) = (4/3)x² - (10/3)x + 1.
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a company finds it can produce 30 heaters for $7500 while producing 40 heaters cost $9000, express the cost y as a linear function of the number of heaters x
express the cost y as a a linear function of the numbers of heatrs x
Answer:
y = 150x + 3000
Step-by-step explanation:
We can express the cost (y) as a linear function of heater quantity (x) by writing the equation in the slope-intercept form (y = mx + b), where m is the slope (in this case change in cost / change in heater quantity) and b is the y-intercept (when x = 0).
To find the slope (m), we can use the slope formula, which is
[tex]m=\frac{y_{2}-y_{1} }{x_{2}-x_{1} }[/tex], where x1 and y1 are one point and x2 and y2 are the other point.
If we allow (30, 7500) to represent x1 and y1 and (40, 9000) to represent x2 and y2, our slope is:
[tex]m=\frac{9000-7500}{40-30}\\ \\m=\frac{1500}{10}\\ \\m=150[/tex]
Now, we can find the y-intercept by plugging in the slope and one of the points and solving for b:
[tex]7500=150(30)+b\\7500=4500+b\\3000=b[/tex]
Thus, our equation where y (cost) is a function of x (heater quantity) is y = 150x + 3000
What is the diameter of a hemisphere with a volume of 62617 cm³, to the nearest tenth of a centimeter?
The diameter of the hemisphere to the nearest tenth of a centimeter is 62.1 cm.
What is hemisphere?
A hemisphere is a three-dimensional geometric shape that consists of a half-sphere, which is a curved surface that is shaped like the surface of a ball, and a flat circular base that lies on a plane perpendicular to the sphere's axis of symmetry. A hemisphere can be thought of as the three-dimensional analogue of a circle.
The volume of a hemisphere is given by the formula:
V = (2/3)π[tex]r^3[/tex]
where V is the volume and r is the radius of the hemisphere.
We are given the volume of the hemisphere as 62617 [tex]cm^3.[/tex]
So, we have:
62617 = (2/3)π[tex]r^3[/tex]
Multiplying both sides by 3/2π, we get:
[tex]r^3[/tex] = 62617 * (3/2π)
Simplifying, we get:
[tex]r^3[/tex] = 29897.4152
Taking the cube root of both sides, we get:
r ≈ 31.04
The diameter of the hemisphere is twice the radius, so:
d = 2r ≈ 62.08
Rounding to the nearest tenth, we get:
d ≈ 62.1
Therefore, the diameter of the hemisphere to the nearest tenth of a centimeter is 62.1 cm.
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Y varies inversley with x. If y = -2 when x = 6, find the value of y when x = -12.
The value of y for the inverse variation when x = -12 is derived to be equal to 1.
What is inverse variationInverse variation is a mathematical relationship between two variables, in which an increase in one variable leads to a proportional decrease in the other variable. Mathematically, inverse variation can be expressed as y = k/x, where y and x are the two variables, k is a constant of proportionality, and the product of y and x is always equal to k.
when x = 6 and y = -2, then k is derived as:
-2 = k/6
k = -12 {cross multiplication}
when x = -12, y is derived as:
y = -12/-12
y = 1
Therefore, the value of y for the inverse variation when x = -12 is derived to be equal to 1.
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A textbook store sold a combined total of 440 math and history textbooks in a week. The number of math textbooks sold was three times the number of history textbooks sold. How many textbooks of each type were sold?
Number of math textbooks is 110 and number of history textbooks is 330 were sold when a textbook store sold a combined total of 440 math and history textbooks in a week.
There are two categories of textbooks. Let "x" stand for the quantity of math textbooks and "y" for the quantity of history textbooks.
A textbook store sold a combined total of 440 math and history textbooks in a week.
Equation,
x + y = 440
The number of math textbooks sold was three times the number of history textbooks sold.
x = 3y
Substituting the values in equation x + y = 440,
3y + y = 440
4y = 440
y = [tex]\frac{440}{4}[/tex]
y = 110
Number of math textbooks = 110
Number of history textbooks:
3y
3 × 110
330
Number of math textbooks is 110 and number of history textbooks is 330 were sold when a textbook store sold a combined total of 440 math and history textbooks in a week.
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complete the square to determine all solutions for x^2-4x=2
Answer: To complete the square, we need to add and subtract the square of half the coefficient of x:
x^2 - 4x = 2
x^2 - 4x + 4 = 2 + 4 (adding and subtracting 4)
(x - 2)^2 = 6
Now we take the square root of both sides:
x - 2 = ±√6
Adding 2 to both sides gives:
x = 2 ± √6
So the solutions are:
x = 2 + √6, 2 - √6
Step-by-step explanation:
Ted invests in an annuity which earns 6.2% annual interest and he contributes $250 per month for 30 years. If Ted wants to perform some calculations to determine his future value, what formula would Ted use?
The future value of Ted's investment after 30 years would be approximately $13,711.86.
What is statistics?
Statistics is a branch of mathematics that deals with the collection, analysis, interpretation, presentation, and organization of numerical data.
Ted can use the formula for the future value of an annuity to calculate the future value of his investment:
FV = Pmt x (([tex](1 + r)^{n}[/tex]) - 1) / r
Where:
FV = Future Value
Pmt = Payment per period (monthly in this case)
r = Interest rate per period (monthly in this case)
n = Number of periods (in this case, 30 x 12 = 360 months)
Substituting the given values, we get:
FV = $250 x (([tex](1 + 0.062/12)^{360}[/tex] - 1) / (0.062/12)
FV = $250 x (283.8576) / 0.0051667
FV = $13,711.86
Therefore, the future value of Ted's investment after 30 years would be approximately $13,711.86.
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Find the slope of a line perpendicular to the line whose equation is 5x + 6y = - 18.
Fully simplify your answer.
Answer:
m = [tex]\frac{6}{5}[/tex]
Step-by-step explanation:
First, we must rewrite 5x + 6y = - 18 to the form y = mx + b.
5x + 6y = -18
6y = -5x - 18
y = [tex]\frac{-5}{ 6}[/tex]x - 3
The equation of a perpendicular line to y = [tex]\frac{-5}{ 6}[/tex]x - 3 must have a slope that is the negative reciprocal of the original slope.
So, the slope of the perpendicular line is
m = [tex]\frac{6}{5}[/tex]
Answer: 6/5
Step-by-step explanation:
trust me
There are some sandwiches on the shelf.
Answer:
367
Step-by-step explanation:
Places in the alphabet:
T=20, H=8, E=5, R=18, E=5. 20+8+5+18+5=56. THERE=56
A=1, R=18, E=5. 1+18+5=24. ARE=24
S=19, O=15, M=13, E=5. 19+15+13+5=52. SOME=52
S=19, A=1, N=14, D=4, W=23, I=9, C=3, H=8, E=5, S=19. 19+1+14+4+23+9+3+8+5+19=105. SANDWICHES=105
O=15, N=14. 15+14=29. ON=29
T=20, H=8, E=5. 20+8+5=33. THE=33
S=19, H=8, E=5, L=12, F=6. 19+8+5+12+6=49. SHELF=49
56+24+52+19+105+29+33+49=367
A swimmer dives into a lake and swims to the surface in a parabolic path. path of the swimmer can be modeled by the equation
h (t) = t - 4t-where h is the height in
feet, and t is the time in seconds.
Complete the statement.
The diver will reach the surface of the water
after seconds.
Answer:
the diver will reach the surface of the water after 1/4 seconds.
Step-by-step explanation:
To find when the swimmer reaches the surface of the water, we need to set h(t) to zero since the surface of the water is at a height of zero feet.
0 = t - 4t^2
Simplifying the equation, we get:
4t^2 = t
4t^2 - t = 0
t(4t - 1) = 0
So, either t = 0 (which corresponds to the initial time when the swimmer is at the surface) or 4t - 1 = 0. Solving for t, we get:
4t - 1 = 0
4t = 1
t = 1/4
Therefore, the diver will reach the surface of the water after 1/4 seconds.
I have an Amazon account that I pay a flat rate of $8 a month. With it, I get free movies and shows. However, any premium movie I purchase costs me $4 more. If I spent $24 last month, how many premium movies did I buy?
The equation would be set up as shown below:
(Let x represent the amount of premium movies I bought.)
4x + 8 = 24
next: subtract 8 from both sides = 4x = 16
then: divide both sides by 4, which gives us our answer = x = 4
So I purchased 4 premium movies.
What are your examples or equations in your day-to-day life
Mr. Campbell has just purchased supplies for his classroom. He bought 76 markers, 31 rulers, and one projector. The markers cost 0.90 each, the rulers 1.60 each, and the projector 97.95 . What was the total amount that he spent?
Answer:
215.95
Step-by-step explanation:
68.4+49.6+97.95
select the correct answer from each drop-down menu. a bag contains five tiles numbered 1 through 5. savannah randomly selects a tile from the bag, replaces the tile, and then draws a second tile. when savannah selects her tiles, selecting the first tile and selecting the second tile are (options on drop down. independent or dependent) events. the probability that the numbers on both tiles are even is (choices on drop down. 20, 40, 10, 16)%.
Selecting the first tile (1st) and selecting the secοnd tile (2nd) are independent events.
The prοbability οf selecting an even number οn a single draw is 2/5 οr 0.4, as there are twο even numbers (2 and 4) οut οf a tοtal οf five pοssible numbers.
Since the events are independent, the prοbability οf selecting an even number οn bοth draws is the prοduct οf the prοbabilities οf selecting an even number οn each draw:
P(bοth even) = P(even οn first) × P(even οn secοnd) = 0.4 × 0.4 = 0.16
Sο the prοbability that the numbers οn bοth tiles are even is 16%.
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Suppose a shipment of 400 components contains 68 defective and 332 non-defective computer components. From the shipment you take a random sample of 25. When sampling with replacement (so that the p = probability of success does not change), note that a success in this case is selecting a defective part. What is the probability of there being no defects in the entire shipment?
The probability of there being no defects in the entire shipment is 0.2075.
What is probability?
Probability can be used to calculate an event's probability. It can only be used to assess the possibility that an event will occur. A scale from 0 to 1, where 0 is impossible and 1 is a particular occurrence.
We are given that out of the 400 components, 68 are defective and 332 are non-defective computer components.
A random sample of 25 is taken.
So,
⇒ Probability (Defects in the shipment) = [tex]\frac{68}{400}[/tex]
⇒ Probability (Defects in the shipment) = 0.17
Thus,
⇒ Probability (No Defects in the shipment) =25 * (1 - 0.17)
⇒ Probability ( No Defects in the shipment) = 25 * (0.83)
⇒ Probability ( No Defects in the shipment) = 0.2075
Hence, the probability of there being no defects in the entire shipment is 0.2075.
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Translate the argument to symbols and use a truth table to determine whether the argument is valid or invalid.
If it rains, then I will watch the Real World marathon.
It did not rain.
.. I did not watch the Real World marathon.
Let p="It rains," and let q="I will watch the Real World marathon."
Part: 0/4
Part 1 of 4
(a) Write the argument in symbols.
Translated into symbols:
If it rains, then I will watch the Real World marathon.
It did not rain.
.. I did not watch the Real World marathon.
The argument to symbols of the logic expression is shown below
Translating the argument to symbolsFrom the question, we have the following parameters that can be used in our computation:
If it rains, then I will watch the Real World marathon.It did not rain.I did not watch the Real World marathon.The argument in symbols can be expressed as:
p -> q
~p
Therefore, ~q
Where:
p = "It rains."
q = "I will watch the Real World marathon."
"->" denotes "implies".
"~" denotes "not".
So the argument can be read as "If it rains, then I will watch the Real World marathon. It did not rain.
Therefore, I did not watch the Real World marathon."
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if correct brainest! look at picture
Answer:
x = 47
Step-by-step explanation:
All angles in a triangle ALWAYS add up to 180 degrees, so to find a missing angle, subtract what is already given. It will be 180 - (102+31), which is the same as 180-133 which equals 47. The basic rule is to subtract given angels from 180 to find the missing side.
Which of the following are the next three terms in the sequence (1, 2, 3, 9, -27, 81, ..}?
The next three terms are:
-243, 729, -2187 rule 1
the output is the square of the input rule 2 the out put is 90 greater than the input
Rule 1: The output is the square of the input. For example, if the input is 3, then the output would be 9 (3 squared).
Rule 2: The output is 90 greater than the input. For example, if the input is 3, then the output would be 93 (3 + 90).
These are the two rules you've provided, and they describe how to calculate the output (y) based on the input (x).
Rule 1: The output is the square of the input.
In this rule, when you have an input value (let's call it x), you square it (multiply it by itself) to get the output value (y). Mathematically, this can be written as:
[tex]y = x^2[/tex]
Rule 2: The output is 90 greater than the input.
In this rule, when you have an input value (x), you add 90 to it to get the output value (y).
Mathematically, this can be written as:
y = x + 90.
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A circular garden has a circumference of 40.25 feet. Find the approximate diameter of the garden. Use 3.14 for π. Round to the nearest hundredth if necessary.
The approximate diameter of the garden is 12.82 feet.
What is circumference?
The circumference of a circle or ellipse in geometry is its perimeter. That is, if the circle were opened up and straightened out to a line segment, the circumference would be the length of the arc. The curve length around any closed figure is more often referred to as the perimeter.
Here, we have
Given: A circular garden has a circumference of 40.25 feet.
We have to find the diameter of the garden.
The formula for the circumference of a circle is given as 2πr.
We have to find the radius of the circular garden
Hence: 2πr = 40.25m
π = 3.14
To find the radius, the formula would be given as
r = C/2π
Where C = Circumference of the circle
r = (40.25/2)x 3.14
r = 63.195
d = C/π
d = 40.25/3.14
d = 12.82
Hence, the approximate diameter of the garden is 12.82 feet.
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Select the correct answer.
Figure 1 is similar to figure 2. What is the height of figure 1?
4x
61-
figure 1
O A.
1 unit
OB.
O C.
O D. 4 units
8 units
2 units
-2x
figure 2
Answer:
D. 4 units
Step-by-step explanation:
Similar figures have corresponding lengths in proportion.
[tex]\frac{4x}{2x}=\frac{x+1}{x} \\ \\ 2=1+\frac{1}{x} \\ \\ 1=\frac{1}{x} \\ \\ x=1[/tex]
Therefore, the height of Figure 1 is 4.
the diameter of each tire on a vehicle is 32 inches. if the tires are moving at a rate of 800 revolutions per minute, find the linear speed of the vehicle in miles per hour.
Answer:
The linear speed of the vehicle is approximately 0.798 miles per hour.
Step-by-step explanation:
The circumference of a tire is given by the formula C = πd, where d is the diameter of the tire. So, the circumference of each tire is:
C = πd = π(32 inches) ≈ 100.53 inches
Now, to find the distance that the vehicle travels in one revolution of each tire, we use the circumference formula again:
distance traveled in one revolution = circumference of the tire = 100.53 inches
Since the tires are rotating at a rate of 800 revolutions per minute, the vehicle travels a distance of:
distance traveled in one minute = distance traveled in one revolution × number of revolutions per minute
= 100.53 inches/revolution × 800 revolutions/minute
= 80424 inches/minute
To convert this to miles per hour, we need to divide by the number of inches per mile and the number of minutes per hour:
speed = (80424 inches/minute) × (1 mile/63360 inches) × (60 minutes/1 hour)
≈ 0.798 miles/hour
Therefore, the linear speed of the vehicle is approximately 0.798 miles per hour.
Select the correct answer. Which expression is equivalent to the given expression? 4 in x + in 3 - in x
Answer:
The expression 4 in x + in 3 - in x can be simplified using logarithmic rules: 4 in x + in 3 - in x = in x^4 + in 3 - in x = in (x^4 * 3 / x) = in (3x^3) Therefore, the expression is equivalent to in (3x^3).
Need to find x using the Pythagorean Theorem
Answer:
1. 15
2. 26
3. ≈ 7.62
4. 8
5. ≈ 23.24
6. √2
7. ≈ 19.42
8. 18
9. x≈10.29 y≈5.83
Step-by-step explanation:
Use a²+b²=c² throughout the problems and plug in the given sides and hypothenuse.
1. 9²+12²=c²
81+144=225
√225 = 15
2. 10²+24²=c²
100+576=676
√676=26
3. 7²+3²=c²
49+9=58
√58 ≈ 7.62
4. a²+6²=10²
a²+36=100
a²=64
x=8
5. a²+6²=24²
a²+36=576
a²=540
x≈ 23.24
6. 1²+1²=c²
1+1=2
x=√2
7. a²+8²=21²
a²+64=441
a²=377
x≈ 19.42
8. a²+24²=30²
a²+576=900
a²=324
x=18
9. for x:
9²+5²=x²
81+25=106
√106 ≈10.29
x ≈10.29
for y:
5²+3²=y²
25+9=34
√34 ≈5.83
y ≈5.83
boom shakalaka
one paycheck is $250 less than another. The sum of the two checks is &1628. how much is each check written for.?
Answer: $939 and $689
Step-by-step explanation:
We can create a system of equations to help us solve. Let x be one and y be the other paycheck.
y = x - $250
y + x = $1,628
Next, we can solve the system of equations by graphing. The point of intersection is our solution. See attached. This gives us (939, 689), so one check was written for $939 and one was written for $689.
help mee with this question
Answer: 22
Step-by-step explanation:
The first one is 10.
The other 2 has 6, because 10-4=6
So
10+(6x2)=22
A company that teaches self-improvement seminars is holding one of its seminars in Wildgrove. The company pays a flat fee of $164 to rent a facility in which to hold each session. Additionally, for every attendee who registers, the company must spend $8 to purchase books and supplies. Each attendee will pay $49 for the seminar. Once a certain number of attendee register, the company will be breaking even. How many attendees will that take? What will be the company's total expenses and revenues?
The company's total expenses and revenues will both be $196 when 4 attendees register.
What is Algebraic expression ?
In mathematics, an algebraic expression is a combination of variables, constants, and mathematical operations, such as addition, subtraction, multiplication, and division, that represents a quantity or a relationship between quantities.
Let's assume that the company's goal is to break even, which means that their total revenue should be equal to their total expenses.
Let's represent the number of attendees by "x." For the company to break even, the total revenue earned from x attendees must be equal to the total expenses incurred.
Total Expenses = Fixed Cost + Variable Cost
Fixed cost = Facility rent fee = $164
Variable cost = Books and supplies per attendee = $8
Total Expenses = $164 + $8x
Total Revenue = Price per attendee x Number of attendees = $49x
To find the break-even point, we set the total revenue equal to the total expenses and solve for x:
$49x = $164 + $8x
$41x = $164
x = 4
Therefore, the company needs to have 4 attendees to break even.
To calculate the total expenses and revenues, we can substitute x = 4 into the equations we derived above:
Total Expenses = $164 + $8(4) = $196
Total Revenue = $49(4) = $196
Therefore, the company's total expenses and revenues will both be $196 when 4 attendees register.
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savanna built a rectangular pen got her goat that was 20 yards long and 15.4 yards wide. what is the area of this pen?
Answer:
308 yards^2
Step-by-step explanation:
Area = length * width
Length = 20
Width = 15.4
20 * 15.4 = 308
Answer: 308 yards
Step-by-step explanation:
One square has a side 3 cm greater than the side of a second square. What are lengths of the sides if the sum of their areas is 1017 cm2. (Make sure to foil your product on the squares)
Using the area formula, the length of the sides of the two squares are 22 cm and 25 cm respectively.
What are squares?A square is a regular quadrilateral in Euclidean geometry, which means that it has four equal sides and four equal angles.
It can alternatively be explained as a rectangle with two neighboring sides that are of equal length.
The square's four sides are equal to one another or congruent.
The square's opposing sides run parallel to one another.
The square's diagonals form a 90° angle with one another.
The square's two diagonals are equal to one another.
Area: s²
s² + (s+3)² = 1017
s² + s² + 9 = 1017
2s² = 1017 - 9
2s² = 1,008
s² = 1008/2
s² = 504
s = ²504
s = 22.44
Round off: 22 cm
Then, (s+3) = 22 + 3 = 25 cm
Therefore using the area formula, the length of the sides of the two squares are 22 cm and 25 cm respectively.
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write your answer using only positive exponents
The value of the expression is 9y⁵z³/x⁶
How to determine the exponentNote that index forms are described as mathematical forms of writing numbers that are large or too small in more convenient forms.
The rules of index forms are;
Add the exponents when the values have the same base variable and are multiplying one another.Subtract the exponents when the values have the same base variable and are dividing one anotherFrom the information given, we have that;
(y³z⁻¹) (3x⁻³y/z⁻²)²
expand the bracket, we get;
(y³z⁻¹) 9x⁻⁶y²/z⁻⁴
Take the inverse exponent of the denominator
(y³z⁻¹) 9x⁻⁶y²z⁴
Now, add the exponents
9x⁻⁶y³⁺²z⁻¹⁺⁴
9x⁻⁶y⁵z³
Then, we have;
9y⁵z³/x⁶
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2. the new principal is $58,254.26. For payment 329, find the following:
a) The interest on $58,254.26
$
b) The payment to principal
$
c) The new principal
$
The interest on $58,254.26 was calculated to be $146.38, the payment to principal was calculated to be $182.62, and the new principal was calculated to be $58,071.64.
What is payment?Payment is the transfer of funds from one party to another in exchange for goods, services, or to fulfill a legal obligation. Payment can take various forms, including cash, check, credit card, money order, or electronic transfer. Payment can be made in person, over the phone, or online. The most important aspect of payment is that it is a legally binding contract between two parties. Payment is an important part of the economic system that facilitates the exchange of goods and services and helps to ensure that all parties involved receive what they expect from the transaction.
The interest on $58,254.26:
To calculate the interest on $58,254.26, we first need to determine the interest rate. Assuming that the interest rate is 3%, we can calculate the interest owed as follows:
Interest = Principal x Interest Rate x Time
Interest = $58,254.26 x 0.03 x 1/12
Interest = $146.38
b) The payment to principal:
To calculate the payment to principal, we need to subtract the interest from the total payment amount.
Payment to Principal = Total Payment – Interest
Payment to Principal = $329 - $146.38
Payment to Principal = $182.62
c) The new principal:
To calculate the new principal, we need to subtract the payment to principal from the original principal.
New Principal = Original Principal – Payment to Principal
New Principal = $58,254.26 - $182.62
New Principal = $58,071.64
In conclusion, the interest on $58,254.26 was calculated to be $146.38, the payment to principal was calculated to be $182.62, and the new principal was calculated to be $58,071.64.
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which number line shows the solotion to one fourth r plus one < five
After looking at the provided number lines, it is clear that the number line (B) accurately reflects 1/4 + 1 < 5.
What is a number line?A number line is a diagram of a graduated straight line used to represent real numbers in introductory mathematics.
It is assumed that every point on a number line corresponds to a real number and that every real number corresponds to a point.
A number on the left of a number line is always smaller than a number on the right.
A number to the right is always greater than a number to the left, too.
So, the inequality we have is:
1/4 + 1 < 5
Now, after observing the given lines, we can easily tell that the number line (B) represents 1/4 + 1 < 5 perfectly.
Therefore, after looking at the provided number lines, it is clear that the number line (B) accurately reflects 1/4 + 1 < 5.
Know more about the number line here:
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Complete question:
Which number line shows the solution to 1/4 + 1 < 5