EXPLANATION
Factors are numbers we can multiply together to get another number.
The factor of 26:
26 divides by 2: 26/2=13
13 divides by 13: 13/13 = 1
2,13 are all prime numbers, therefore no further factorization is possible.
Add the primer factors:
2,13
Add 1 and the number 26 itself:
1,26
The factors of 26 are 1,2,13,26.
How many quarts of pure antifreeze must be added to 3 quarts of a 50% antifreeze solution to obtain a 60% antifreeze solution?quart(s) of pure antifreeze must be added(Round to the nearest tenth as needed.)
Initially we have 3 quarts of a 50% antifreeze solution
We want to obtain a 60% antifreeze solution by adding x quarts of pure antifreeze
Therefore we can set the following equation,
[tex]1x+0.5*3=0.6*(x+3)[/tex]where x are the quarts of pure antifreeze added
let's solve for x
[tex]\begin{gathered} x+1.5=0.6x+1.8 \\ x-0.6x=1.8-1.5 \\ 0.4x=0.3 \\ x=\frac{0.3}{0.4} \\ x=0.75 \end{gathered}[/tex]rounding to the nearest tenth, 0.8 quarts of pure antifreeze must be added
factor the equation 3x^2 +16x-12
(3x - 2)(x + 6)
Explanation:3x² +16x-12
a = 3, b = 16, c = - 12
a(c) = 3(-12) = -36
The factors of -36 whose sum will give +16 are -2 and 18
3x² + 18x - 2x - 12
3x(x + 6) -2(x + 6)
(3x - 2)(x + 6)
The factorisation: (3x - 2)(x + 6)
The price of a toy is increased by 20%. The resulting price is later decreased by $40.00. If the original price of the toy is $60.00, what is the final price of the toy ?
Answer
Final price of the toy = $32.00
Explanation
The price of a toy is increased by 20%
The resulting price is then decreased by $40.00
Original Price = $60.00
The price of a toy is increased by 20%
Resulting price after this increase = 1.2 (60) = $72
The resulting price is then decreased by $40.00
Final Price = 72 - 40 = $32.00
Hope this Helps!!!
harlyne deposited $3400 into a savings account that has an annual simple interest rate of 0.2%. Find the amount in the savings account after each number of years.
2 years $
5 years $
8 years $
The simple interest after 2, 5 and 8 years on a principal of $3400 at 0.2% are $3413.6, $3434 and $3454.4 respectively
SIMPLE INTERESTSimple interest is a quick and easy method to calculate interest on the money, in the simple interest method interest always applies to the original principal amount, with the same rate of interest for every time cycle.
Simple interest is calculated with the following formula: S.I. = P × R × T, where P = Principal, R = Rate of Interest in % per annum, and T = Time, usually calculated as the number of years. The rate of interest is in percentage r(%) and is to be written as r/100.
Principal: The principal is the amount that initially borrowed from the bank or invested. The principal is denoted by P.Rate: Rate is the rate of interest at which the principal amount is given to someone for a certain time, the rate of interest can be 5%, 10%, or 13%, etc. The rate of interest is denoted by R.Time: Time is the duration for which the principal amount is given to someone. Time is denoted by T.Amount: When a person takes a loan from a bank, he/she has to return the principal borrowed plus the interest amount, and this total returned is called Amount.Using the data given;
S.I = P × R × T
When T = 2 years
S.I = 3400 * 0.002 * 2
S.I = 13.6
The amount after 2 years = 3400 + 13. 6 = $3413.6
When T = 5 years
S.I = 3400 * 0.002 * 5
S.I = 34
The amount after 5 years = 34 + 3400 = $3434
When T = 8 years
S.I = $3454.4
Learn more on simple interest here;
https://brainly.com/question/3575751
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Go step by step to reduce the radical. V 243 DVD try You must answer all questions above in order to submit.
We are given the following radical
[tex]\sqrt[]{243}[/tex]Let us reduce the above radical
We need to break the number 243 into a product of factors
Notice that 81 and 3 are the factors (83×3 = 243)
[tex]\sqrt[]{243}=\sqrt[]{81}\cdot\sqrt[]{3}[/tex]Since 81 is a perfect square so the radical becomes
[tex]\sqrt[]{243}=\sqrt[]{81}\cdot\sqrt[]{3}=9\cdot\sqrt[]{3}[/tex]Therefore, the simplified radical is
[tex]undefined[/tex]wpn Learning. UIC 3. Solve by elimination. x + 2y = -7 x - 5y = 7 A. (-7,0) B. (-3, -2) C. (-2,-3) D. (0, -7)
x+2y=-7 ------> equation 1
x-5y=7 -------->equation 2
Change the signs in equation 2
x+2y=-7 ------> equation 1
-x+5y=-7 -------->equation 2
Add equation 1 and 2
x+2y=-7 ------> equation 1
-x+5y=-7 -------->equation 2
_________
7y=-14
y=-14/7
y=-2
Now substitute y=-2 in equation 1,
x+2(-2)=-7
x-4=-7
x=-7+4
x=-3
(x,y)=(-3,-2)
Option B is the correct answer.
i inserted a picture of the question which is question 7, i can give you the answer to the previous question which is question 6 if it helps
The first thing we have to know is that the trinomial (polynomial of three terms) is called a perfect square trinomial, the polynomial that when factorized gives us a perfect squared expression in the following way
[tex]a^2+2ab+b^2=(a+b)^2[/tex][tex]\begin{gathered} w^2-3w=350 \\ a=1 \\ 2ab=-3 \\ b=-\frac{3}{2} \\ b^2=\frac{9}{4} \end{gathered}[/tex][tex]\begin{gathered} w^2-3w-\frac{9}{4}=350-\frac{9}{4} \\ w^2-3w-\frac{9}{4}=347.75 \\ (w-\frac{3}{2})^2=347.75 \end{gathered}[/tex][tex](w-\frac{3}{2})^2-\frac{1409}{4}=0\to\text{answer}[/tex]Please help me on my hw I need help on #2
Given:
The number is,
[tex]5.232323\ldots\text{.}[/tex]To express the given number into fraction . it means in the form,
[tex]\frac{a}{b}[/tex]We can express the given number into geometric series as,
[tex]\begin{gathered} 5.232323\ldots=5+\frac{23}{100}+\frac{23}{10000}+\frac{23}{100000}+\text{.}\ldots\ldots \\ =5+\frac{23}{100}+23(\frac{1}{100})^2+23(\frac{1}{100})^3+\text{.}\ldots\ldots\text{.}\mathrm{}(1) \\ \frac{23}{100}+23(\frac{1}{100})^2+23(\frac{1}{100})^3+\text{.}\ldots=-23+\sum ^{\infty}_{n\mathop=1}23(\frac{1}{100})^{n-1} \\ =-23+\frac{23}{1-\frac{1}{100}} \\ =-23+\frac{23(100)}{99} \\ =-23+\frac{2300}{99} \\ =\frac{-2277+2300}{99} \\ =\frac{23}{99} \end{gathered}[/tex]Now, equation (1) becomes,
[tex]5+\frac{23}{99}=\frac{518}{99}[/tex]Answer:
[tex]\frac{518}{99}[/tex]Determine if the following equations are parallel, perpendicular, or neither. 7(x – 1) = 3y + 21 and 3.5x + 1.5y = 4.5
Given the two equations
7(x - 1) = 3y + 21
3.5x + 1.5y = 4.5
To determine if the lines are visible or perpendicular
Step 1: Expand the equations and make y the subject of the formula
7(x - 1) = 3y + 21
=> 7x - 7 = 3y + 21
=> 3y = 7x - 7 - 21
=> 3y = 7x -28
Divide both sides by 3
y = 7/3x - 28/3 ------equation 1
3.5x + 1.5y = 4.5
1.5y = -3.5x + 4.5
Divide both sides by 1.5
y = -3.5/1.5 x + 4.5/1.5
y = -7/3x + 3--------equation 2
Step 2: compare the two equations to the equation of a line, y = mx + c
For equation 1
m = 7/3
For equation 2
m= -7/3
The slopes are not the same
Also, the product of the gradients did not give -1
It can be seen that the lines are neither perpendicular nor parallel
The plastic lid of a cylindrical container is a circle. The lid has a radius of 9centimeters. What is the circumference of the lid?
We are asked to determine the circumference of a 9 cm radius circle. To do that we will use the following formula:
[tex]S=2\pi r[/tex]Where:
[tex]\begin{gathered} S=\text{ circumference} \\ r=\text{ radius} \end{gathered}[/tex]Now, we substitute the value of the radius:
[tex]S=2\pi(9cm)[/tex]Solving the operations:
[tex]S=18\pi=56.55cm[/tex]Therefore, the circumference is 56.55 cm.
⦁ It takes the earth 24 h to complete a full rotation. It takes Mercury approximately 58 days, 15 h, and 30 min to complete a full rotation. How many hours does it take Mercury to complete a full rotation? Show your work using the correct conversion factors.
Answer:
Answer:
58 days, 15 h, and 30 min
Step-by-step explanation:
any letters or words in your answer! Ciera works at a day care center. Her job is to make sure there are always enough adult workers. Last month, there were 7 adults for 56 children, which is the minimum ratio allowed under state law. This month, 74 children are expected to enroll. How many adults will there need to be at the center? Your answer
10 adults
Explanation
Step 1
find the unit rate
[tex]\text{unit rate=}\frac{Number\text{ of Children}}{\text{Number of adults}}[/tex]Let
number of children=56
number of adults=7
replace,
[tex]\begin{gathered} \text{unit rate=}\frac{Number\text{ of Children}}{\text{Number of adults}}= \\ \text{unit rate=}\frac{56\text{ Children}}{7\text{ adults}} \\ \text{unit rate= 8 Children per adult} \end{gathered}[/tex]Step 2
Let x represents the adults needed for 74 children
find the unit rate:
[tex]\begin{gathered} \text{unit rate=}\frac{Number\text{ of Children}}{\text{Number of adults}} \\ \text{unit rate=}\frac{74}{x} \end{gathered}[/tex]as the unit rate must be the same
[tex]\begin{gathered} 8\frac{Children}{\text{adult}}=\frac{74}{x} \\ 8=\frac{74}{x} \\ cross\text{ multiply} \\ 8x=74\cdot1 \\ 8x=74 \\ \text{divide both sides by 8} \\ \frac{8x}{8}=\frac{74}{8} \\ x=9.25 \\ \text{hence, we n}eed\text{ 10 adults at the center} \end{gathered}[/tex]Which sequence of transformations maps polygon ABCD onto polygon WXYZ?
We have to find the transformations that led from polygon ABCD to WXYZ.
As the shapes are not equally oriented, we have to find if one of the transformation is a rotation or a reflection.
We can fin this by looking at the position of corresponding sides. So first, we have to find corresponding sides of the two polygons. The polygon WXYZ has also a scale transformation, so its size is proportional, with a proportion greater than 1 as it is bigger, to the size of ABCD.
Each side in the pre-image has a corresponding side in the image. Each corresponding side in the image will be k times bigger than the side in the pre-image, and this k is the same for the four sides.
We can look at the sides that are parallel to the axis, BC and CD, and see that CD is longer than BC. If we look at WXYZ, YZ is longer than YX.
Then, we can conclude that YZ and CD are corresponding sides as BC and YX.
The scale factor is k = 2 as YZ is twice as long as CD.
Then we can see, by the position of BC and CD respect to YX and YZ that no rotation can convert the pre-image into the image, so the orientation of the image is due to a reflection with axis of symmetry at x = 7.
Then, after the reflection, the image is dilated with a factor k = 2.
Answer:
B. A reflection of polygon ABCD followed by a dilation of the image with a scale factor of 2.
Answer:
B
Step-by-step explanation:
plato
Correctnomial function with the stated properties. Reduce all fractions to lowest terms.Third-degree, with zeros of - 3, - 1, and 2, and passes through the point (3, 5).
We must construct a polynomial with the following characteristics:
0. degree: 3,
,1. zeros: x₁ = -3, x₂ = -1 and x₃ = 2,
,2. passes through the point (3, 5).
The general form for this polynomial is:
[tex]p(x)=a*(x-x_1)(x-x_2)(x_{}_{}-x_3).[/tex]Where a is a constant factor and x₁, x₂ and x₃ are the zeros of the polynomial.
Replacing the values of the zeros, we have:
[tex]p(x)=a*(x+3)(x+1)(x-2).[/tex]Using the condition that the polynomial passes through (3, 5), we have:
[tex]y=a*(3+3)(3+1)(3-2)=a*24=5.[/tex]Solving for a, we get a = 5/24. Replacing this value in the equation above, we get:
[tex]p(x)=\frac{5}{24}(x+3)(x+1)(x-2).[/tex]Answer[tex]p(x)=\frac{5}{24}(x+3)(x+1)(x-2)[/tex]a line segment has the endpoint T(2,4) and the midpoint of (3,6.5). find the coordinates of the other point B.
a line segment has the endpoint T(2,4) and the midpoint of (3,6.5). find the coordinates of the other point B.
we know that
The formula to calculate the midpoint between two points is equal to
[tex]M(\frac{x1+x2}{2},\frac{y1+y2}{2})[/tex]In this problem we have
M=(3,6.5)
(x1,y1)=T(2,4)
(x2,y2)=B
substitute the given values in the formula above
[tex](3,6.5)=(\frac{2+x2}{2},\frac{4+y2}{2})[/tex]Find the value of x2 coordinate
3=(2+x2)/2
x2+2=6
x2=6-2=4
Find the value of y2 coordinate
6.5=(4+y2)/2
y2+4=13
y2=13-4=9
therefore
the coordinates of point B(4,9)
es? Explain your reasoning.
35. MP MODELING REAL LIFE A planet orbiting a star
at a distance such that its temperatures are right for
liquid water is said to be in the star's habitable zone. The
habitable zone of a particular star is at least 0.023 AU
and at most 0.054 AU from the star (1 AU is equal to the
distance between Earth and the Sun). Draw a graph tha
represents the habitable zone.
the distanco
♥it must orbit in a zone where liquid water is possible♥
Find remaining zereos of f
the function is:
degree: 4
and we know that some zeros are:
[tex]x=7i,3,-3[/tex]because the function is degree 4 we know that it should have 4 zeros. we also know that the function is symetric because of the real solutions, so the las solution will be the negative of the imaginari root so the remaining zeros are:
[tex]x=-7i[/tex]Find the speed of each train (set up a table )
We have two trains that leave Mexico City at the same time, one to the east and the other to the west. The train that travels to the west is 10 mph slower than the other train. The diagram of the problem is:
The combined velocity is:
[tex]V_T=v+v-4=2v-4[/tex]The equation that relates the distance D between the trains and the time after t hours is:
[tex]D=V_T\cdot t[/tex]If after 1.5 hours the trains are 171 miles apart, then using the equation above:
[tex]\begin{gathered} 171=V_T\cdot1.5 \\ V_T=\frac{171}{1.5} \\ V_T=114 \\ 2v-4=114 \\ 2v=118 \\ v=59\text{ mph} \end{gathered}[/tex]Now, the speeds are:
[tex]\begin{gathered} \text{East}\colon59\text{ mph} \\ \text{West}\colon55\text{ mph} \end{gathered}[/tex]Craig earns $2.50 per hour plus $3.50 for each haircut he gives.He worked 7 hours and gave 4 haircuts. How much did he earn?a. $31.50b. $168c. $46d. $17.50
Craig earns $2.50 per hour plus $3.50 for each hair cut he gives.
He worked for 7 hours.
In hour basis, he earned
[tex]2.50\times7=17.5[/tex]He gave 4 hair cuts.
For 4 haircuts, he earned
[tex]3.50\times4=14[/tex]So, he total earned
[tex]17.5+14=31.5[/tex]Hence, the correct option is (A).
The radius of a quarter circle is 3 millimeters. What is the quarter circle's perimete r=3 mm ude 3.14 for .. millimeters Submit can you explain
Given:
The radius of the quarter circle is given 3 mm.
To find:
The perimeter of the quarter circle.
Solution:
It is known that the perimeter of the quarter circle is given by:
[tex]2r+\frac{\pi r}{2}[/tex]So, the perimeter of the quarter circle:
[tex]\begin{gathered} P=2r+\frac{\pi r}{2} \\ =2(3)+\frac{3.14\times3}{2} \\ =6+4.71 \\ =10.71 \end{gathered}[/tex]Thus, the perimeter of the quarter circle is 10.71 mm.
State the null and alternative hypotheses for the claimThe average score of high school basketball games is less than 88.
Convert: 3 days = minutes
ANSWER
4320 minutes
EXPLANATION
To convert from days to minutes, first, we have to convert from days to hours. It is known that 1 day has 24 hours, so 3 days have,
[tex]3\text{ }days\cdot\frac{24\text{ }hours}{1\text{ }day}=72\text{ }hours[/tex]Then, we convert from hours to minutes. If 1 hour has 60 minutes,
[tex]72\text{ }hours\cdot\frac{60\text{ }minutes}{1\text{ }hour}=4320\text{ }minutes[/tex]Hence, there are 4320 minutes in 3 days.
The board of directors of a company must have select a president, a secretary and a treasurer in how many possible ways can this be accomplished if there are 22 members on the board
Given
Total number of members = 22
Find
Possible ways of selection of president, a secretary and a treasurer
Explanation
As we know , the number of possible ways of selection is given by
[tex]N=^nP_r[/tex]there are three members required so , r = 3
now , substitute the values in above equation
[tex]\begin{gathered} N=^{22}P_3 \\ N=\frac{22!}{(22-3)!} \\ \\ N=\frac{22!}{19!} \\ \\ N=22\times21\times20 \\ N=9240 \end{gathered}[/tex]Final Answer
Possible ways of selection of president, a secretary and a treasurer = 9240
The units of the subway map below are in miles. Suppose the routes between stations are straight. Find the approximate distance a passenger would travel between stations J and K.
Point J has coordinates (2,6)
Point K has coordinates (-1,-3)
The distance between 2 point is given by
[tex]d=\sqrt[]{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]where
[tex]\begin{gathered} (x_1,y_1)=\mleft(2,6\mright) \\ (x_2,y_2)=(-1,-3) \end{gathered}[/tex]By substituying these values, we have
[tex]\begin{gathered} d=\sqrt[]{(-1_{}-2)^2+(-3-6)^2} \\ d=\sqrt[]{(-3)^2+(-9)^2} \\ d=\sqrt[]{9+81} \\ d=\sqrt[]{90} \end{gathered}[/tex]hence,
[tex]d=9.49[/tex]Find Sec A and Cot B exactly if a=8 and b=7
The given triangle is right angle triangle with side a = 8 and b 7
Apply pythagoras theorem for the side c;
In a right-angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides.
Hypotenuse² = Perpendicular² + Base²
Here, perpendicular, a =8 and Base b = 7
[tex]\begin{gathered} c^{2}=a^{2}+b^{2} \\ c^{2}=8^{2}+7^{2} \\ c^2=113 \\ c=\sqrt[]{113} \end{gathered}[/tex]The trignometric ratio for sec of an angle define as the ratio of the hypotenuse to the side adjacent to a given angle in a right triangle.
[tex]\begin{gathered} \sec A=\frac{Hypotenuse}{Adjacent\text{ side of angle A}} \\ \sec A=\frac{c}{b} \\ \sec A=\frac{\sqrt[]{113}}{7} \end{gathered}[/tex]The trignometric ratio for cosine of angle define as the ratio of the adjacent side to the the opposite side of the angle,
[tex]\begin{gathered} CotB=\frac{\text{Adjacent side to angle B}}{\text{Opposite side to angle B}} \\ CotB=\frac{a}{b} \\ CotB=\frac{8}{7} \end{gathered}[/tex]Answer; a)
[tex]\text{SecA}=\frac{\sqrt[]{113}}{7},\cot B=\frac{8}{7}[/tex]
4. Suppose that you receive a movie-rental bill for the month that is much higher thanit usually is. Currently you are paying $3.99 for each movie you rent. Switching to asubscription would allow you to watch unlimited movies for only $7.99 per month. However,during a normal month you don't have much time to sit and watch movies. You do not reallywant to waste your money on a monthly subscription. You decide to check your onlinebilling statements and make a probability distribution for the number of movies you mightwatch each month.The results are in the following table:Number of Movies, X 0 1 2 3 4 5Probability, P(X) 0.10 0.15 ? 0.35 0.14 0.13a) What is the probability you will watch 2 movies next month, i.e., P(X=2)?b) What is the probability that you will watch more than 2 movies next month, i.e., P(X<4) orP(X<=3)?c) How much would you expect to spend, on a per-month basis, should you continue to pay foreach movie separately? Explain your answer
For the given table representing the probability distribution, the probability of watching 2 movies is unknown.
The sum of all probabilities should be equal to 1. We can use that to calculate the unknown probability:
Adding all probabilities and equating to 1:
[tex]0.1+0.15+P(x=2)+0.35+0.14+0.13=1[/tex]Solving for P(x=2)
[tex]\begin{gathered} 0.87+P(x=2)=1 \\ P(x=2)=1-0.87 \\ P(x=2)=0.13 \end{gathered}[/tex]Then: A. The probability of watching 2 movies next month is 0.13.
The complete table of probability distribution will look like this:
xP(x)
00.1
10.15
20.13
30.35
40.14
50.13
To calculate the probability of watching more than two movies we need to add the probabilities of watching 3, 4 or 5 movies. Those should be added because those are the cases where more than 2 movies are watched.
[tex]\begin{gathered} P(x>2)=P(x=3)+P(x=4)+P(x=5) \\ P(x>2)=0.35+0.14+0.13 \\ P(x>2)=0.62 \end{gathered}[/tex]Then, B. The probability of watching more than 2 movies is 0.62.
The probability of watching 2 movies is equivalent to P(x>2) or P(x>=3).
On the other hand, to calculate P(X<4) or P(X<=3) we need to add the probabilities of watching 3 movies or less. That is, probabilities of watching 0, 1, 2 or 3:
[tex]\begin{gathered} P(x<4)\text{ or }P(x\le3)=P(x=0)+P(x=1)+P(x=2)+P(x=3) \\ P(x<4)\text{ or }P(x\le3)=0.1+0.15+0.13+0.35 \\ P(x<4)\text{ or }P(x\le3)=0.73 \end{gathered}[/tex]The probability of watching 3 movies or less next month is 0.73.
To estimate how much we would expect to spend per month if we pay for each movie sepparately we need to calculate the expected value of movies per month.
We can estimate that with the probability distribution given in the table.
The expected value is the sum of the products between each event and their probabilities:
[tex]\text{Expected Value}=\sum ^{}_{}x\cdot P(x)[/tex]Let's call EV the expected value:
[tex]\begin{gathered} EV=(0\cdot0.1)+(1\cdot0.15)+(2\cdot0.13)+(3\cdot0.35)+(4\cdot0.14)+(5\cdot0.13) \\ EV=2.67 \end{gathered}[/tex]Then, we should expect to watch about 2.67 movies per month, on average.
I each individual movie costs $3.99, then, the total expenses per month will be:
[tex]2.67\cdot3.99\approx10.65[/tex]Then, C. According to the given probability distribution, we should expect to watch about 2.67 movies per month, on average, and spend in total $10.65 per month. We would be spending more than we would if we selected the unlimited movies plan which costs only $7.99 per month, then, it would be wise to decide to change our subscription to that plan.
what is the perimeter? and what unit should i use?
The given information is:
- 7-gon: it has 7 sides.
- a=15 ft
- s=14 ft
The perimeter, is the sum of the side lengths, as it has 7 sides, then its perimeter is:
[tex]\begin{gathered} P=7s \\ P=7*14ft \\ P=98ft \end{gathered}[/tex]Now, the area is given by the formula:
[tex]\begin{gathered} A=\frac{7}{2}(s*a) \\ \\ A=\frac{7}{2}(14ft*15ft) \\ \\ A=\frac{7}{2}(210ft^2) \\ \\ A=\frac{1470ft^2}{2} \\ \\ A=735ft^2 \end{gathered}[/tex]The area is 735 square feet
Instructions: Factor the polynomial expression.7x²-13x +6Answer:
Explanation
Given the polynomial
[tex]7x^2-13x+6[/tex]We can find its factors below.
[tex]\begin{gathered} 7x^2-13x+6 \\ =7x^2-7x-6x+6 \\ =7x(x-1)-6(x-1) \\ =(7x-6)(x-1) \end{gathered}[/tex]Answer: (7x-6) and (x-1)
Solve the system of two linear inequalities graphically.ſr<62-3Step 2 of 3 : Graph the solution set of the second linear inequality.AnswerKeyboThe line will be drawn once all required data is provided and will update whenever a value is updated. The regions will be added once the line is drawn.Enable Zoom/PanChoose the type of boundary line:Solid (-) Dashed (-)Enter two points on the boundary line:510-35Select the region you wish to be shaded:
Answer and Explanation:
Given the system of two linear inequalities;
[tex]\begin{gathered} x<6 \\ x\ge-3 \end{gathered}[/tex]To solve the above system of inequalities graphically, we follow the below steps;
Step 1: Graph the first inequality;
Since the first inequality has a less than sign, we'll shade the region to the left of the line.
Also, the first inequality does not have an equality sign, so the line will be a dashed line.
See below the graph of the first inequality;
Step 2: Graph the second inequality on the same grid;
Since the inequality has an equality sign, the line will be a solid line.
Also, the inequality has the greater than sign, so we'll shade the region to the right of the line
See below the image of the graph;
Step 3: The solution set of the two systems of inequalities is the region where the shading overlaps.
As can be seen in the above graph, the shaded region between the dashed line and the solid line is the solution of the system of inequalities.
The graph of the solution set of the second inequality is as shown below;
On the boundary line we can select the below points;
[tex]\begin{gathered} \lparen-3,5) \\ \lparen-3,-5) \end{gathered}[/tex]Solve for x in the equation x2+2x+ 1 = 17.X=-1+ /15X=-1+ /17X=-2+2.15X=-1+ /13
First, write the quadratic equation in standard form. Then, use the quadratic formula to find the solutions for the quadratic equation.
Remember that if a quadratic equation is written in standard form:
[tex]ax^2+bx+c=0[/tex]Where a, b and c are constants, then the solutions for x are given by:
[tex]x=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a}[/tex]Starting with the given equation:
[tex]x^2+2x+1=17[/tex]Substract 17 from both members to write the equation in standard form:
[tex]\begin{gathered} \Rightarrow x^2+2x+1-17=17-17 \\ \Rightarrow x^2+2x-16=0 \end{gathered}[/tex]Use the quadratic formula, setting a=1, b=2 and c=-16:
[tex]\begin{gathered} x=\frac{-(2)\pm\sqrt[]{(2)^2-4(1)(-16)}}{2(1)} \\ =\frac{-2\pm\sqrt[]{4+64}}{2} \\ =\frac{-2\pm\sqrt[]{68}}{2} \end{gathered}[/tex]Simplify the expression using the properties of radicals. Since 68 is equal to 4 times 17, then:
[tex]\begin{gathered} x=\frac{-2\pm\sqrt[]{68}}{2} \\ =\frac{-2\pm\sqrt[]{4\cdot17}}{2} \\ =\frac{-2\pm\sqrt[]{4}\cdot\sqrt[]{17}}{2} \\ =\frac{-2\pm2\cdot\sqrt[]{17}}{2} \\ =\frac{2(-1\pm\sqrt[]{17})}{2} \\ =-1\pm\sqrt[]{17} \end{gathered}[/tex]Therefore, the solutions for x in the given equation are:
[tex]\begin{gathered} x_1=-1+\sqrt[]{17} \\ x_2=-1-\sqrt[]{17} \end{gathered}[/tex]