Answers
The quadratic polynomial with integer as coefficients that has four real roots including these conjugates is a² + b² - 2ab - 20a - 20b + 88 ≥ 0
Here is how to derive the polynomialOne possible quartic polynomial with integer coefficients that has four real roots, including the conjugates 5+√3 and 5-√3, is:
(x - 5 - √3)(x - 5 + √3)(x - a)(x - b)
where a and b are the remaining two roots.
Expanding the first two factors using the difference of squares formula, we get:
(x - 5)² - 3
Expanding the last two factors, we get:
x² - (a + b)x + ab
Putting it all together, the quartic polynomial is:
a² + b² - 2ab - 20a - 20b + 88 ≥ 0
To ensure that a and b are also real, we can use the fact that the discriminant of a quadratic equation ax² + bx + c = 0 with real coefficients is b² - 4ac ≥ 0. Applying this to the quadratic factor in the above polynomial, we get:
(a + b)² - 4ab ≥ 0
Expanding and simplifying, we get:
a² + b² - 2ab - 20a - 20b + 88 ≥ 0
To simplify the problem, we can set a = b, since the polynomial is symmetric with respect to the roots. Then, the inequality becomes:
2a² - 40a + 88 ≥ 0
Dividing by 2 and factoring, we get:
(a - 10)² - 12 ≥ 0
This inequality is always true for real values of a, so we can choose any real value for a and set b = a to obtain a quartic polynomial with integer coefficients that has four real roots, including the conjugates 5+√3 and 5-√3.
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Related Questions
Christina Sanders is concerned about the financing of a home. She saw a small Cape Cod–style house that sells for $90,000. If she puts 10% down, what will her monthly payment be at (a) 30 years, 5%; (b) 30 years, 512% ; (c) 30 years, 6%; and (d) 15 years, 378% ? What is the total cost of interest over the cost of the loan for each assumption? (Use Table 15.1.) Note: Round your answers to the nearest cent. Do not round intermediate calculations.
Answers
The monthly payments and total cost of interest depend on the interest rate and the length of the loan. Generally, a lower interest rate or a shorter loan term results in lower total interest costs.
To calculate the monthly payments, we can use the formula for the present value of an annuity:
[tex]PV = P * (1 - (1 + r)^{-n}) / r[/tex]
where PV is the present value of the loan, P is the principal (the amount borrowed), r is the monthly interest rate, and n is the number of monthly payments.
(a) 30 years, 5%:
PV = 0.9 * $90,000 = $81,000 (since she puts 10% down)
r = 0.05 / 12 = 0.0125(monthly interest rate)
n = 30 * 12 = 360 (number of monthly payments)
Using the formula, we get:
PV = $81,000
Monthly payment = $436.82
Total cost of interest = $57,255.85
(b) 30 years, 5.12%:
r = 0.0512 / 12 = 0.00427 (monthly interest rate)
Using the formula, we get:
PV = $81,000
Monthly payment = $439.88
Total cost of interest = $59,956.19
(c) 30 years, 6%:
r = 0.06 / 12 = 0.005 (monthly interest rate)
Using the formula, we get:
PV = $81,000
Monthly payment = $485.24
Total cost of interest = $95,687.34
(d) 15 years, 3.78%:
n = 15 * 12 = 180 (number of monthly payments)
r = 0.0378 / 12 = 0.00315 (monthly interest rate)
Using the formula, we get:
PV = $81,000
Monthly payment = $591.93
Total cost of interest = $34,147.07
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I need help with these
Answers
19. __C__ Segment QC is the radius (1/2 diameter)
20. __K__ Arc BDE is major because three points are listed
21. __E__ Segment AD is the diameter (length cutting the circle in half)
22. __L__ Arc ABD has two endpoints on the circle’s diameter
23. __H__ Angle AQB has an apex at the circle’s center, Q
24. __A__ Point Q is the circle’s center
25. __D__ Segment ED is a chord as it has endpoints on the circular arc
26. __B__ Point C is perpendicular to the radius (QC) where it meets the circle.
27. __F__ Segment AB is a secant because it contains a chord and intersects the circle at 2 points.
28. __J__ Arc EA is minor between two points
29. __G__ Segment CG is the tangent as it intersects the circle at one point, C.
Hope this helps!
what is the volume of a hemisphere with a diameter of 8ft (rounded to the nearest tenth of a cubic foot)
Answers
well, if the hemisphere has a diameter of 8, that means its radius is half that or 4, hmm let's find the volume of a Sphere with such a radius, then let's halve it.
[tex]\textit{volume of a sphere}\\\\ V=\cfrac{4\pi r^3}{3}~~ \begin{cases} r=radius\\[-0.5em] \hrulefill\\ r=4 \end{cases}\implies V=\cfrac{4\pi (4)^3}{3} \\\\\\ \stackrel{\textit{now let's get half of that for the hemisphere}}{V=\cfrac{1}{2}\cdot \cfrac{4\pi (4)^3}{3}\implies V=\cfrac{128\pi }{3}}\implies \boxed{V\approx 134.0}[/tex]
PLS HELP LITERALLY HELP HELPL
Answers
[tex]-37 < -12v+1 \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ v=-11\hspace{5em}-37 < -12(-11)+1\implies -37 < 133 ~~ \checkmark \\\\[-0.35em] ~\dotfill\\\\ v=-9\hspace{5em}-37 < -12(-9)+1\implies -37 < 109~~ \checkmark \\\\[-0.35em] ~\dotfill\\\\ v=-5\hspace{5em}-37 < -12(-5)+1\implies -37 < 61~~ \checkmark \\\\[-0.35em] ~\dotfill\\\\ v=6\hspace{5em}-37 < -12(6)+1\implies -37 < -71 ~~ \bigotimes[/tex]
3x+9-8x=-31?????????????
Answers
3x + 9 - 8x = -31
Combining like terms, we get:
-5x + 9 = -31
Subtracting 9 from both sides, we get:
-5x = -40
Dividing both sides by -5, we get:
x = 8
Therefore, the solution is x = 8. We cannot simplify this any further.
72+z rewrite the expression as a produc 72
Answers
Rewriting the expression based on the information will be 72(1 + z/72).
How to explain the expressionIn order to transform 72 + z into a product based on 72, we can apply the distributive property of multiplication over addition. Essentially, this property maintains that:
a x (b + c) = a x b + a x c
By extension, we can create the following equation:
72 + z = 72 + 1 x z
In order to produce an equivalent expression as a multiple of 72 and another factor, we would then need to use the distributive rule once again:
72 + 1 x z = 72 x 1 + 72 x z/72
Simplification leads us to arrive at:
72 + z = 72(1 + z/72)
Our end result is expressed as a synthesis between 72 and the expression (1 + z/72), namely:
72(1 + z/72).
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Question 2
Plot the coordinates on the number line provided and then find the coordinate of the indicated point
on a number line that partitions the segment into the given ratio.
(The numbers are not aligned properly to the marks, so redraw the number line.)
A is at -2 and B is at 14. Find the point, T, so that T is three-fourths of the distance from A to B.
-10
5
10
Answers
Note that in the line graph presented, 3/4 the distance from A to B is 12, thus, t = 12.
what is the explanation about this?
Given:
A (- 2 , 0) and B(14, 0)
Using distance formula, we can determine the distance between coordinates given:
d = √ ( (x2 - x1)² + ( y2 - y1) ²)
In this case, x1 = -2 y1 = 0,
x2 = 14, and y2 = 0.
Substituting these values into the formula, we get:
d = √( (14 - (-2 ) ) ² + (0 - 0) ²)
= √((16)² + (0)² )
= √(256)
= 16
The distanc between the points (-2,0) and (14,0) is 16 units.
Recall that we need to find the position of t which is 3/4 the distance between A and B.
That is:
3/4 x 16
t = 12.
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if i have 7 shirts and 2 pants how many outfits can i make
Answers
Answer:14
Step-by-step explanation:
2 pants can only go with the different shirts once. So it should be 14
A rectangular fish tank 35 cm long, 30 cm wide and 25 cm high is filled with water to a depth of 20 cm.
Find the volume of water in the fish tank.
Give your answer in liters.
Answers
Answer:
First, we need to calculate the volume of the rectangular fish tank. The volume of a rectangular solid is given by the formula:
volume = length x width x height
Plugging in the values we have, we get:
volume = 35 cm x 30 cm x 25 cm = 26,250 cubic cm
Next, we need to find the volume of water in the fish tank. The volume of water is equal to the volume of the space the water occupies, which is a rectangular solid with length 35 cm, width 30 cm, and height 20 cm. The volume of this rectangular solid is:
volume = length x width x height = 35 cm x 30 cm x 20 cm = 21,000 cubic cm
To convert cubic centimeters (cm^3) to liters (L), we need to divide by 1000:
volume = 21,000 cm^3 ÷ 1000 = 21 L
Therefore, the volume of water in the fish tank is 21 liters.
Step-by-step explanation:
What will be the answer?
Answers
A symbolic representation of the contrapositive is
letter D
Answer:
b
Step-by-step explanation:
What function is represented in the table?
Select one:
y = 3(2^x)
y = 2(3^x)
y = 2(.5^x)
y = 3^x
Answers
The exponential function represented in the table is given as follows:
y = 3^x.
How to define an exponential function?An exponential function has the definition presented as follows:
y = ab^x.
In which the parameters are given as follows:
a is the value of y when x = 0.b is the rate of change.From the table, we have that when x= 0, y = 1, hence the parameter a is given as follows:
a = 1.
When x is increased by one, y is multiplied by three, hence the parameter b is given as follows:
b = 3.
Hence the function is:
y = 3^x.
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Tomas used 3 [tex]\frac{1}{2}[/tex] of flour and now has 1 and two-thirds cups left. Which equation can he use to find f, the number of cups of flour he had to begin with?
f + 3 and one-third = 1 and two-thirds
f minus 3 and one-third = 1 and two-thirds
3 and one-third f = 1 and two-thirds
f divided by 3 and one-third = 1 and two-thirds
Answers
Answer:
B
Step-by-step explanation:
Answer:
I think It's b or a
Step-by-step explanation:
did this like 4 weeks ago.
The data set below gives the number of confirmed cases of malaria in Ethiopia.
Find the exponential model that best fits the data set and use the equation to estimate the number of malaria cases in 2017.
Year 2001 2002 2005 2007 2009 2010 2012
Malaria 392 428 539 451 1036 1158 1693
cases
In thousands
Answers
The number of malaria cases in 2017 is 2700.
Let the dependent variable, y is malaria cases in thousands and the independent variable, x is the year.
Here, x is the number of years starting in the year 2000.
The required model is obtained as [tex]y = 299e^{0.1295x}[/tex]
Now, the predicted number of malaria cases in the year 2017, that is, x = 17 years can be computed as,
[tex]y = 299e^{0.1295(17)}\\\\= 299(9.020\\\\= 2700[/tex]
Hence, the number of malaria cases in 2017 is 2700.
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A survey asked people of different ages whether they get their news by
reading the paper. What is the probability that a person surveyed is 40 or
above, given that he or she gets the news by reading the paper? If necessary,
round your answer to the nearest percent.
Don't read
Read paper
Total
Under 40
40 or older
Total
4
24
28
paper
36
16
52
40
40
80
Answers
The probability that a person surveyed is 40 or older and gets the news by reading the paper is 60%. The Option C is correct.
How do we find the reading for 40 or older?We must find the probability of a person getting their news by reading the paper and are given with a condition that the person must be 40 or older.
Data
The survey is conducted among 80 people.
Number of people aged 40 or older = 40
No of people aged 40 or older and read paper = 24
The probability of a person aged 40 or older reads a paper will be:
= (No. of People aged 40 or older and read paper)/(No. of people aged 40 or older)
= 24/40
= 0.6
= 60%
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Which function is a translation one unit right of the function f(x) = log x?
h(x) = log x + 1
g(x) = log x-1
j(x) = log(x + 1)
k(x) = log(x - 1)
Answers
Answer:
The function that is a translation one unit right of the function f(x) = log x is k(x) = log(x - 1).
When we shift a function f(x) one unit right, we replace x with (x - a) where "a" is the amount of the shift. In this case, "a" is equal to 1. Therefore, the function becomes:
f(x - 1) = log(x - 1)
This means that the function k(x) = log(x - 1) is obtained by shifting the function f(x) one unit to the right.
pleas anwear fast
Graph the function.
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(
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)
=
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⋅
(
3
2
)
�
f(x)=4⋅(
2
3
)
x
f, left parenthesis, x, right parenthesis, equals, 4, dot, left parenthesis, start fraction, 3, divided by, 2, end fraction, right parenthesis, start superscript, x, end superscript
Answers
The graph of the function f(x) = 4(2/3)^x is added as an attachment
How to determine the graph of the functionFrom the question, we have the following parameters that can be used in our computation:
f(x)=4⋅(
2
3
)
x
Express the equation properly
So, we have
f(x) = 4(2/3)^x
The above expression is a an equation of a exponential function
Next, we plot the graph using a graphing tool
To plot the graph, we enter the equation in a graphing tool and attach the display
See attachment for the graph of the function
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PLEASE HELP (WILL GIVE BRAINLIEST
Answers
The exact volume of the soup can is 18.375 π cubic inches. So, correct option is C.
The volume of a cylinder can be found by multiplying the area of the base (circle) by the height of the cylinder. In this case, the diameter of the base is 3.5 inches, which means the radius is 1.75 inches (since radius = diameter/2).
So the area of the base can be calculated as:
π × (1.75 in)² = π × 3.0625 in²
The height of the cylinder is given as 6 inches.
Therefore, the exact volume of the soup can would be:
π × (1.75 in)² × 6 in = π × 3.0625 in² × 6 in = 18.375 π cubic inches (rounded to two decimal places)
So, correct option is C.
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How many times can u write your name in 12 minutes using variables Based on your recorded values, write an equation that represents the relationship between the time you spent doing the activity and the amount of the activity you completed. Make sure to define your variables
:D
Answers
The following equation denotes the correlation between the duration and output of an activity: A = kT
How to explain the equationIt should be noted that A represents the amount of production, T is the period of working on the task, and k consists of a constant ratio.
Therefore, it can be conclusively stated that the culminating yield is in direct correlation to the amount of effort contributed, as denoted by the factor of proportionality k, thus producing a linear relationship when displayed upon a graph.
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Which compares the end behavior of the functions f and g?
f(x) = −17x − 9 g(x) = − 78
7
8
x + 20
A. For function f, as x → ∞
x
→
∞
, f(x) → ∞
f
(
x
)
→
∞
. Likewise, for function g, as x → ∞
x
→
∞
, g(x) → ∞
g
(
x
)
→
∞
.
B. For function f, as x → ∞
x
→
∞
, f(x) → −∞
f
(
x
)
→
−
∞
. Likewise, for function g, as x → ∞
x
→
∞
, g(x) → −∞
g
(
x
)
→
−
∞
.
C. For function f, as x → ∞
x
→
∞
, f(x) → −∞
f
(
x
)
→
-
∞
. However, for function g, as x → ∞
x
→
∞
, g(x) → ∞
g
(
x
)
→
∞
.
D. For function f, as x → ∞
x
→
∞
, f(x) → ∞
f
(
x
)
→
∞
. However, for function g, as x → ∞
x
→
∞
, g(x) → −∞
g
(
x
)
→
−
∞
.
Answers
Answer:
The correct option that compares the end behavior of the functions f and g is D.
For function f, as x → ∞, f(x) → -∞, which means that the function approaches negative infinity as x approaches infinity. This is because the leading term of the function is -17x, which approaches negative infinity as x approaches infinity.
For function g, as x → ∞, g(x) → -∞, which means that the function also approaches negative infinity as x approaches infinity. This is because the leading term of the function is -78/87x, which approaches negative infinity as x approaches infinity.
Therefore, both functions have the same end behavior, which is approaching negative infinity as x approaches infinity.
Help help help hehehheheheh
Answers
Answer:
Step-by-step explanation:
your supposed to learn things in class pay attetion
A lab technician has to fill the tank given below.
Find its volume.
Use
π=3.14 and round your answer to the nearest hundredth.
Answers
Answer:
π(5^2)(8) + (1/2)(4/3)π(5^3)
= 200π + 250π/3 = 850π/3 cubic meters
= about 889.67 cubic meters
HELP 15 POINTS
Solve for x.
X
12 cm
x = [?] cm
Round to the nearest hundredth.
620
Answers
The opposite side of the right triangle is determined as 10.6 cm.
What is the value of x?The value of x is calculated by applying trigonometry ratio as shown below;
we will use the short form; SOH CAH TOA
SOH = sin θ = opposite /hypothenuse side
TOA = tan θ = opposite side / adjacent side
CAH = cos θ = adjacent side / hypothenuse side
The hypotenuse side of the right triangle is given, and we are required to find the opposite side of the right triangle.
sin (62) = x/12
x = 12 sin (62)
x = 10.6 cm
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PLEASE HELP ILL DO SNYTHING PLSPLSPLS
5. Beth is 63 years old. She loves to bike. She decides to look at other clubs to join. Why do you think she didn’t want to join this club? Explain your answer using the stem-and-leaf plot above. (1 point)
Answers
Beth could not join the club because the highest age which is allowed here is only 55 years
Since, A statistical expression obtained from a list of data that refers an abnormal gap from other values.
The statistical rules that instruct us to divides the data or observation values into four parts.
After analyzing the stem leaf diagram, we noticed the youngest age allowed to join the club is 10 years and the club allow highest 55 years old to join.
the plot also defines that the number of members is declined with the age. There is only person in the club who is 55 years. There are few members with the age over 40 years.
hence, Beth did not join the club because her age was above the highest age that allowed in this club.
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Find the volume of rectangular prism. (You should have at least 2 steps shows in your work!)(show all decimal places or a fraction)
Answers
The volume of this rectangular prism include the following: 84.375 cubic meters.
How to calculate the volume of a rectangular prism?In Mathematics and Geometry, the volume of a rectangular prism can be calculated by using the following formula:
Volume of a rectangular prism = L × W × H
Where:
L represents the length of a rectangular prism.W represents the width of a rectangular prism.H represents the height of a rectangular prism.By substituting the given dimensions (parameters) into the formula for the volume of a rectangular prism, we have the following;
Volume of rectangular prism = 5 × 4 1/2 × 3 3/4
Volume of rectangular prism = 5 × 4.5 × 3.75
Volume of rectangular prism = 84.375 cubic meters.
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Marques is going to invest in an account paying an interest rate of 2% compounded
daily. How much would Marques need to invest, to the nearest dollar, for the value of
the account to reach $28,000 in 6 years?
Answers
In this case, we have:
A = $28,000
r = 2% = 0.02
n = 365 (since interest is compounded daily)
t = 6
We want to solve for P. Substituting the given values into the formula, we get:
$28,000 = P * (1 + 0.02/365)^(365*6)
Dividing both sides by (1 + 0.02/365)^(365*6), we get:
P = $28,000 / (1 + 0.02/365)^(365*6) = $22,406.57
Therefore, Marques would need to invest $22,407 (rounded to the nearest dollar) for the value of the account to reach $28,000 in 6 years.
1. Out of 318 seventh and eighth grade CAVA students there were 48 students that chose Music as their elective
course and the rest chose World Language. There were a total of 124 eighth grade students and 108 of them
chose World Language as their elective.
Use this information to complete the two-way table completely.
Enter your answer by filling in the boxes to complete the table (5 pts)
Answer:
8th Grade
7th Grade
Totals
Music
48
World Language
108
Totals
124
318
Answers
The students in 8th class who choose music are 16 in number,
The students in 7th class who choose music are 32 in number
The total who choose world language is 270
The totals who are in 7th grade be 194
The number of students who choose world language in 7th class are 162
Let x be the students in 8th class who choose music
x+108=124
x=124-108
x=16 students
Now y be the students in 7th class who choose music
16+y=48
y=48-16
y=32
Let z be the total who choose world language
48+z=318
z=318-48
z=270
Now the totals who are in 7th grade be A
124+A=318
A=318-124
A=194
B be the number of students who choose world language in 7th class
Now 32+B=194
B=194-32
B=162
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Describe by a write up on how models can be used to express numbers of different bases to base 2 and base 5
Answers
Conversion models can be used to express numbers of different bases to base-2 and base-5. To convert a number from base-10 to base-2, we use division-by-2, and for base-5, we divide by 5. To convert between bases, we convert to base-10 first and then to the desired base. for example 25 in binary is 11001, 45 in base-5 is 104, 89 to base-2 is 1011001.
In computer science and mathematics, binary (base-2) and quinary (base-5) numbering systems are commonly used to represent numbers. However, it is often necessary to express numbers in other bases as well. This is where conversion models come into play.
To convert a number from base-10 to base-2, we can use the division-by-2 method.
The process involves repeatedly dividing the decimal number by 2 and keeping track of the remainders until the quotient becomes 0. We then write the remainders in reverse order to get the binary representation of the number.
For example, to convert the number 25 to binary
25 / 2 = 12 R1
12 / 2 = 6 R0
6 / 2 = 3 R0
3 / 2 = 1 R1
1 / 2 = 0 R1
Therefore, 25 in binary is 11001.
To convert a number from base-10 to base-5, we can use a similar method. Instead of dividing by 2, we divide by 5 and keep track of the remainders. For example, to convert the number 45 to base-5
45 / 5 = 9 R0
9 / 5 = 1 R4
1 / 5 = 0 R1
Therefore, 45 in base-5 is 104.
We can also convert a number from base-2 to base-5 or vice versa. To convert a number from base-2 to base-5, we can first convert it to base-10 and then to base-5. For example, to convert the binary number 101110 to base-5
1*2⁵ + 0*2⁴ + 1*2³ + 1*2² + 1*2¹ + 0*2⁰ = 46
46 to base-5 is 141
To convert a number from base-5 to base-2, we can first convert it to base-10 and then to base-2. For example, to convert the quinary number 324 to base-2
3*5² + 2*5¹ + 4*5⁰ = 89
89 to base-2 is 1011001
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Mark bought 55 lbs of clay for his art projects. He used 12.4 lbs to make a sculpture, and 0.75 lbs for each mug. How many mugs did Mark make if he had 32.85 lbs of clay left over?
Answers
If Mark used 12.4 lbs to make a sculpture, and 0.75 lbs for each mug then the number of mugs he can make is 57.
Mark bought 55 lbs of clay for his art projects
He used 12.4 lbs to make a sculpture
The left over clay is 55-12.4 = 42.6
We know that 0.75 lbs for each mug
We have to find the number of mugs he can make from left over clay
To find this we have to divide 42.6 by 0.75
Number of mugs = 42.6/0.75
=56.8
=57
Hence, if Mark used 12.4 lbs to make a sculpture, and 0.75 lbs for each mug then the number of mugs he can make is 57.
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Find the area of the trapezoid at the right 6 cm base 5 cm heights
Answers
Which number line shows a graph of the inequality x < 6
Answers
The number line that shows the inequality x < 6 is given by the following option:
Option D.
What are the inequality symbols?The four inequality symbols, along with their meaning on the number line and the coordinate plane, are presented as follows:
> x: the amount is greater than x -> the number is to the right of x with an open dot at the number line. -> points above the dashed horizontal line y = x on the coordinate plane.< x: the amount is less than x. -> the number is to the left of x with an open dot at the number line. -> points below the dashed horizontal line y = x on the coordinate plane.≥ x: the amount is at least x. -> the number is to the right of x with a closed dot at the number line. -> points above the solid vertical line y = x on the coordinate plane.≤ the amount is at most x. -> the number is to the left of x with a closed dot at the number line. -> points above the dashed vertical line y = x on the coordinate plane.The inequality for this problem is given as follows:
x < 6.
Which is composed by the numbers to the left of x = 6, with an open circle, due to the open interval.
More can be learned about inequalities at brainly.com/question/25275758
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