Answers
Answer:
The solutions to this equation are x = -8/9 and x = -16/3.
Step-by-step explanation:
We have this equation here:
[tex]\displaystyle \frac{1}{3}|x-4|=\frac{2}{3} x+2|x+\frac{6}{3} |[/tex]First, let's simplify the right sight of the equation by simplifying 6/3 to 2.
[tex]\displaystyle \frac{1}{3}|x-4|=\frac{2}{3} x+2|x+2 |[/tex]Multiply both sides of the equation by 3 in order to get rid of the fractions.
[tex]|x-4|=2x+6|x+2|[/tex]Move the terms on either side of the equation to set them equal to 0.
[tex]|x-4|-2x-6|x+2|=0[/tex]Now, we can split this equation up into 4 possible cases. In each case, we make the absolute values negative or positive.
Case #1 (first absolute value: positive; second absolute value: positive)[tex](x-4)-2x-6(x+2)=0[/tex]Simplify this equation by distributing -6 inside the parentheses.
[tex]x-4-2x-6x-12=0[/tex]Combine like terms.
[tex]-7x-16=0[/tex]Add 16 to both sides of the equation and divide by -7.
[tex]\displaystyle \boxed{x=-\frac{16}{7}}[/tex]Case #2 (first absolute value: negative; second absolute value: positive)[tex]-(x-4)-2x-6(x+2)=0[/tex]Distribute the negative sign and -6 inside of their respective parentheses.
[tex]-x+4-2x-6x-12=0[/tex]Combine like terms.
[tex]-9x-8=0[/tex]Add 8 to both sides of the equation and divide by -9.
[tex]\displaystyle \boxed{x=-\frac{8}{9}}[/tex]Case #3 (first absolute value: positive; second absolute value: negative)[tex](x-4)-2x-6[-(x+2)]=0[/tex]Distribute the negative sign inside the parentheses first.
[tex]x-4-2x-6(-x-2)=0[/tex]Now, distribute -6 inside the parentheses.
[tex]x-4-2x+6x+12=0[/tex]Combine like terms.
[tex]5x+8=0[/tex]Subtract 8 from both sides of the equation and divide by 5.
[tex]\displaystyle \boxed{ x=-\frac{8}{5}}[/tex]Case #4 (first absolute value: negative; second absolute value: negative)[tex]-(x-4)-2x-6[-(x+2)]=0[/tex]Distribute the negative signs inside the parentheses first.
[tex]-x+4-2x-6(-x-2)=0[/tex]Distribute -6 inside the parentheses.
[tex]-x+4-2x+6x+12=0[/tex]Combine like terms.
[tex]3x+16=0[/tex]Subtract 16 from both sides of the equation and divide by 3.
[tex]\displaystyle \boxed{x=-\frac{16}{3}}[/tex]Extraneous SolutionsWhenever we solve problems with absolute values, we will always need to check for extraneous solutions.
Definition: These are solutions that may come up while solving but do not actually fit in the domain of the original problem.
Checking for these is tedious, but it will help eliminate wrong answers, so let's plug every "solution" for x that we found back into the original equation.
1) x = -16/7Substitute this value back into the original equation.
[tex]\displaystyle \frac{1}{3}|(-\frac{16}{7}) -4|=\frac{2}{3} (-\frac{16}{7}) +2|(-\frac{16}{7}) +\frac{6}{3} |[/tex]If we do all the calculations correctly, we will get:
[tex]\displaystyle \frac{44}{21} =-\frac{20}{21}[/tex]Since this equation is NOT true, this means that x = -16/7 is NOT A SOLUTION.
2) x = -8/9Substitute this value back into the original equation.
[tex]\displaystyle \frac{1}{3}|(-\frac{8}{9}) -4|=\frac{2}{3} (-\frac{8}{9}) +2|(-\frac{8}{9}) +\frac{6}{3} |[/tex]If we do all the calculations correctly, we will get:
[tex]\displaystyle \frac{44}{27} =\frac{44}{27}[/tex]Since this equation IS true, this means that x = -8/9 IS A SOLUTION.
3) x = -8/5Substitute this value back into the original equation.
[tex]\displaystyle \frac{1}{3}|(-\frac{8}{5}) -4|=\frac{2}{3} (-\frac{8}{5}) +2|(-\frac{8}{5}) +\frac{6}{3} |[/tex]If we do all the calculations correctly, we will get:
[tex]\displaystyle \frac{28}{15} =-\frac{4}{15}[/tex]Since this equation is NOT true, this means that x = -8/5 is NOT A SOLUTION.
4) x = -16/3Substitute this value back into the original equation.
[tex]\displaystyle \frac{1}{3}|(-\frac{16}{3}) -4|=\frac{2}{3} (-\frac{16}{3}) +2|(-\frac{16}{3}) +\frac{6}{3} |[/tex]If we do all the calculations correctly, we will get:
[tex]\displaystyle \frac{28}{9} =\frac{28}{9}[/tex]Since this equation IS true, this means that x = -16/3 IS A SOLUTION.
Final AnswerThe two true solutions of the double absolute value equation, shown above, are:
[tex]\displaystyle \boxed{-\frac{8}{9} } \ \ \& \ \ \boxed{-\frac{16}{3}}[/tex]Answer:
x = -16/3 or -8/9
Step-by-step explanation:
The equation can be rewritten as a piecewise linear function with two breakpoints, or three domain regions. Each breakpoint is at the value of x where the argument of the absolute value function is zero: at x=4 and x=-2.
For each absolute value, we have ...
|q| = -q for q < 0
|q| = q for q ≥ 0
Then the three parts of the domain are ...
x < -2 . . . . . . . . where |x +2| has its vertex-2 ≤ x < 4 . . . . . between the vertices4 ≤ x . . . . . . . . . where |x -4| has its vertex__
Subtracting the right side, the equation becomes ...
1/3|x -4| -2/3x -2|x +2| = 0
Then the three piecewise functions are ...
x < -2Both absolute value arguments are negated.
1/3(-(x -4)) -2/3x -2(-(x +2)) = 0
x(-1/3 -2/3 +2) +4/3 +4 = 0 . . . . . collect terms
x +5 1/3 = 0 . . . . . . . simplify
x = -5 1/3 . . . . . . . . . subtract 5 1/3. This result is in the domain
__
-2 ≤ x < 4Only the argument of |x -4| is negated.
1/3(-(x -4)) -2/3x -2(x +2) = 0
x(-1/3 -2/3 -2) +4/3 -4 = 0 . . . . collect terms
-3x -8/3 = 0 . . . . . . . simplify
-3x = 8/3 . . . . . . . add 8/3
x = -8/9 . . . . . divide by -3. This result is in the domain
__
4 ≤ xNeither absolute value function argument is negated.
1/3(x -4) -2/3x -2(x +2) = 0
x(1/3 -2/3 -2) -4/3 -4 = 0 . . . . . collect terms
-7/3x -8/3 = 0 . . . . . . . . . simplify
-7/3x = 8/3 . . . . . . . . add 8/3
x = -8/7 . . . . . . . . . divide by -7/3. This result is not in the domain.
There is no solution in this region.
__
The solutions are ...
x = -5 1/3x = -8/9
Related Questions
The mean of four numbers is 59, The numbers are 52, 58, 57, and x. Find the value of x.
Answers
Answer:
X=69
Step-by-step explanation:
52, 58, 57, X =59
52+58+57+X/4=59
167+X/4=59
Multiply both sides by 4
4(167+X) /4 = 59x4
Simplify
167 + X= 236
Subtract 167 from both sides
167 + X- 167 = 236 – 167
Simplify
X= 69
to check
52+ 58+57+69/4 =59
If RS= 5 , RT = 12 , ST = 7 , and points R, S, and T are collinear, what is the relationship between points R , S , and T ?
Answers
The relationship between points R , S , and T is that the sum of the RS and RT will be equal to RT that is RT=RS+ST
What are collinear points?Collinear points are defined as the points which lie on the same line. Here we have three points R, S and T that are collinear.
We have the length equal to:
RS=5, RT=12 ST=7
So we can set the relationship as the sum of the RS and RT will be equal to RT.
So the relation will be given as:-
RT=RS+ST
hence the relationship between points R , S , and T is that the sum of the RS and RT will be equal to RT that is RT=RS+ST
To know more about collinear points follow
https://brainly.com/question/13071707
#SPJ1
Answer:
R, S, and T
Step-by-step explanation:
collinear, is the relationship between all the points
d. Your doctor tells you you are in the 30th percentile for blood pressure among US adults. What is your systolic BP? Round to 2 decimal places.
Answers
Using the normal distribution, it is found that your systolic blood pressure is of 110.45.
Normal Probability DistributionThe z-score of a measure X of a normally distributed variable with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex] is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The z-score measures how many standard deviations the measure is above or below the mean. Looking at the z-score table, the p-value associated with this z-score is found, which is the percentile of X.Researching the problem on the internet, the mean and the standard deviation are given, respectively, by:
[tex]\mu = 122, \sigma = 22[/tex].
The 30th percentile is X when Z = -0.525, hence your BP is given as follows.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]-0.525 = \frac{X - 122}{22}[/tex]
X - 122 = -0.525 x 22
X = 110.45.
More can be learned about the normal distribution at https://brainly.com/question/24663213
#SPJ1
In a recent survey of gun control laws, a random sample of 1000 women showed that 65% were in favor of stricter gun control laws. In a random sample of 1000 men, 60% favored stricter gun control laws. Construct a 95% confidence interval for p1 − p2.
Answers
Answer:
[tex]CI\approx\{0.0076,0.0924\}[/tex]
Step-by-step explanation:
The formula for a z-confidence interval for a difference of proportions is [tex]\displaystyle CI=(\hat{p}_1-\hat{p}_2)\pm z^*\sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1}+\frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}[/tex]:
Let [tex]\hat{p}_1=0.65[/tex] to represent the sample proportion of [tex]n_1=1000[/tex] women who are in favor of stricter gun control laws and let [tex]\hat{p}_2=0.60[/tex] to represent the sample proportion of [tex]n_2=1000[/tex] men who are in favor of stricter gun control laws.
A 95% confidence level corresponds to a critical value of [tex]z^*=1.96[/tex], so now we perform the necessary calculations assuming conditions are met:
[tex]\displaystyle CI=(0.65-0.60)\pm 1.96\sqrt{\frac{0.65(1-0.65)}{1000}+\frac{0.60(1-0.60)}{1000}}\approx\{0.0076,0.0924\}[/tex]
Hence, we are 95% confident that the true difference in the proportion of women and men who favor stricter gun control laws is between 0.0076 and 0.0924
Since the interval does not contain "0" and only contain values greater than "0", it is more likely that the proportion of women who were in favor of stricter gun control laws is higher than the proportion of men who were in favor of stricter gun control laws.
I need help with this problem please.
Answers
Answer:
D
Step-by-step explanation:
What is the measure of angle x?
Answers
Answer:
90 degrees cuz the lil square thang
Answer:
x = 48°
Step-by-step explanation:
The sum of angles on a line is 180°.
The angles 42°, 90°, and x° lie on the same line.
42 + 90 + x = 180x + 132 = 180x = 48°Jill bought 3 pints of strawberries for $3.98 each. What is the best estimate of how much Jill spent?
$4.00
$9.00
$12.00
$8.00
Answers
Answer:
Answer C. $12.00
Step One: Round
First you will want to round $3.98 to the nearst hundredth. A little saying my teacher taught me to help you to remember to round:
Find your place, and move next door.
Five or over add one more!
Numbers to the left, stay the same.
Numbers to the right, zeros your name!
So, we choose the 8 in 3.98. We then round. Since the nine is over five, we add one more to it. Causing it to be 4.00
Step Two: Multiply
Now multiply buy 3 since Jill is getting three pints of strawberries
3 times 4 is twelve
So, your answer is 12.00
Hope this helps!!
Given a triangle ABC at points A=(-4,-3) B=(4, 5) C=(2,-4), and if the triangle is
dilated with a scale factor of 3, centered at the origin, find the new point C'.
Answers
Answer: (6,-12)
Step-by-step explanation:
For dilations centered at the origin, we can multiply each of the coordinates by the scale factor. This means that [tex]C'=(2 \times 3, -4 \times 3)=\boxed{(6, -12)}[/tex]
Solve |x|+7<4.
A.) {x|x<-11 or x>-3}
B.) ∅
C.) {x|-3
Answers
Answer:
No solutions
Step-by-step explanation:
Given inequality,
|x|+7<4To Find:
Value of x in the inequality.
Solution:
Add -7 to both sides
= |x|+7+(−7)<4+(−7)= |x|<−3Find absolute value
No solutionsWhy?
Reason:Because, absolute values can't be less than 0.
Factor completely.
5x² + 25x + 20 =
Answers
Answer: 5(x + 1)(x + 4)
Step-by-step explanation Tell me if it is wrong
Find the greatest common factor.
7x³a +7x²a²
Enter the correct answer.
Answers
Answer:
7x²a(x + a)
Step-by-step explanation:
7x³a = 7 * x * x * x * a
7x²a² = 7 *x * x * a *a
Common factor = 7x²a
So we have to take the common factor from the two terms
7x³a + 7x²a² = 7x²a ( x + a)
There are 22 fourth-grade students and 26 fifth-grade students who became
cheerleaders. Each cheerleading group will have 8 members. Which equation
can be used to find out how many groups can be made?
Answers
(22 + 26) / 8 = x
(/ this means division)
Please solve (17 points)
Answers
Answer:
50.6
Step-by-step explanation:
Hola has 18 balls and he wants to give some of his balls to 9 girls on the bed. If he gave away some balls how many balls are there left?
Answers
Answer:
the answer should be 2
Step-by-step explanation:
you have to divide the 18 balls by the 9 girls.
Identify the intercepts of the graph.
Answers
Answer: X-ints = (-1,0) and Y-ints = (1,0)
Step-by-step explanation:
This is quite simple. The way to solve this problem is by visualization, if you have learned how coordinates are created, this is how you would do it.
In a 2D plane, you have an X and Y axis and coordinates are created in the form of (X,Y).
by simply looking at the graph, I can see how the line crosses the positive 1 in the Y-axis, and -1 in the X-axis.
Part A) asks to find the X-intercept(s) which as stated above, it croses -1 on the X-axis and 0 on the Y-axis. Therefore the intercept is:
(-1,0)
part B) asks the same but for the Y-intercepts. We look at the graph and see the line crosses positive 1 in the Y-axis, and 0 in the X-axis. So:
(0,1)
done :)
rewrite the equation in Ax+By=C form. use intergers for A,B,C
y=1/5-1
Answers
Answer:
[tex]5y=-4[/tex]
Sure hope this helps you :D
When the common ratio of an
infinite geometric sequence is
less than 1, its sum can be
found by the formula
a
where s is the sum, a is the first
term, and r is the common
ratio, What is the sum of the
geometric sequence below?
16,8,4,2.
Answers
The sum of the infinite geometric sequence 16, 8, 4, 2 is 32
How to find the sum of a geometric sequence?The sum can be represented as follows:
s∞ = a / 1 - r
where
r = common ratio
a = first term
Therefore,
r = 8 / 16 = 1 / 2
a = 16
hence,
s = 16 / 1 - 1 / 2
s = 16 / 1 / 2
s = 16 × 2 / 1
s = 32
learn more on geometric sequence here: https://brainly.com/question/2288956
#SPJ1
How can you tell whether two graph functions
represent inverse functions or not?
Answers
Answer:
If is not one to one is not inverse (no inverse)
Step-by-step explanation:
o is the centre of the circle find the size of each angle
Answers
Answer:
In the figure,
AC is the diameter of circle with centre O
∠DAC=37
o
AD || BC
∠ACB=∠DAC (Alternate angles)
Hence,
x=37
o
In Δ ABC,
∠B=90
o
(angle in semicircle)
By angle sum property of triangle, we get,
∠x+∠y+∠B=180
o
37
o
+∠y+90
o
=180
o
∠y=180
o
−127
o
We get,
∠y=53
o
Step-by-step explanation:
Two arithmetic means between 3 and 24 are -
Answers
The value of two arithmetic means which are inserted between 3 and 24 are 24/9 and 75/9.
What is arithmetic mean?Arithmetic mean is the mean or average which is equal to the ratio of sum of all the group numbers to the total numbers.
The two arithmetic means between 3 and 24 are has to be inserted.
3, A₂, A₃, 24
All the four numbers are in arithmetic progression. The nth terms of AM can be found using the following formula:
t(n)=a(n-1)d
Here, d is the common difference a is the first terms and n is the total term. The first term, a=3 and t₄=24. Thus, the common difference is;
[tex]t(n)=a(n-1)d\\t(4)=3(4-1)d\\24=3(3)d\\24=9d\\d=\dfrac{24}{9}[/tex]
The second and 3rd term are:
[tex]A₂=3+\dfrac{24}{9}=\dfrac{51}{9}\\ A₃=\dfrac{51}{9}+\dfrac{24}{9}=\dfrac{75}{9}[/tex]
Thus, the value of two arithmetic means which are inserted between 3 and 24 are 24/9 and 75/9.
Learn more about the arithmetic mean here;
https://brainly.com/question/14831274
#SPJ1
now bsf yall know........
Answers
Answer:
24 - 14.4b
Step-by-step explanation:
percentage are boys? 3. The price of a chocolate bar is 25p and includes 5p profit. Express the profit as a percentage of the original price.
Answers
Answer:
25%
Step-by-step explanation:
the original price is 20 not 25 because 25 is the total of the profit and the price of the chocolate so there's no way 25 is the original price and if you know that and you know the formula it'll be easy for you to get the percentage profit good day.
16 out of 80 baseball players are left handed.
What percent of the players are left handed?
Answers
Answer:
20% of players are left-handed
Step-by-step explanation:
First, find how much each player is worth percent-wise
100 / 80 = 1.25
Then find out how much percent of players that makes up
1.25 x 16 = 20
The answer is 20% of players are left-handed, to check it you could do
80 * 0.2 = 16
(80 = total players, 0.2 = 20%, 16 = total left handed players)
3f–4g=1
6f–6g=5
use the elimination method
Answers
6f-6g= 5 —(2)
(1)x2:6f-8g= 2 —(1a)
(1a)-(2):(6f-8g)-(6f-6g)= 2-5
6f-8g-6f+6g= -3
-2g= -3
g= 3/2
Substitute g= 3/2 into (1):
3f-4(3/2)=1
3f-6=1
3f=7
f =7/3
Multiple Choice Find AD and ED for parallelogram ABCD.
A) AD = 13
B) ED = 13
C) ED=9
D) ED = 16
E) AD = 9
F) AD = 16
Answers
Answer: A, C
Step-by-step explanation:
Diagonals of a parallelogram bisect each other, so [tex]4x-7=2x+1 \longrightarrow x=4[/tex]. So [tex]ED=4(4)-7=\boxed{9}[/tex].
Opposite sides of a parallelogram are congruent, so [tex]8y-3=3y+7 \longrightarrow y=2[/tex], so [tex]AD=8(2)-3=\boxed{13}[/tex]
Which statement correctly compares the shapes of the distributions?
Answers
The statement that compares the distribution shapes is (b) East Hills HS is positively skewed, and Southview HS is symmetric.
How to interpret the distribution shapes?From the question, we have the following highlights:
East Hills HS has more points at the right. This represents a positive skewSouthview HS has more points at the center. This represents a symmetric skewnessHence, the statement that compares the distribution shapes is (b) East Hills HS is positively skewed, and Southview HS is symmetric.
Read more about skewness at:
https://brainly.com/question/24309209
#SPJ1
Answer:
Step-by-step explanation:
East hills is negative skewed, Southview is symmetric
Explanation-just took the quiz
Look at the image below.
3
2.2
2
What is the area of the parallelogram?
square units
Answers
Please help me my friends
What is the perimeter of this rectangle?
Answers
The length and width of the rectangle will be √68 and √17. Then the perimeter of the rectangle will be 2 (√17 + √68).
What is a rectangle?It is a polygon with four sides. The total interior angle is 360 degrees. A rectangle's opposite sides are parallel and equal, and each angle is 90 degrees. Its diagonals are all the same length and intersect in the center.
The graph is shown below.
Then the perimeter of the rectangle will be
P = 2 (length + width)
The length of the rectangle will be
L² = 2² + 8²
L² = 4 + 64
L² = 68
L = √68
The width of the rectangle will be
W² = 1² + 4²
W² = 1 + 16
W² = 17
W = √17
Then the perimeter will be
P = 2 (√17 + √68)
More about the rectangle link is given below.
https://brainly.com/question/10046743
#SPJ1
It costs $3.00 to rent a boat on the lake for hour. Select ALL the statements that correctly match this ratio.
4
3
- of a hour costs $9.00
4
2 hours costs $18.00
1 hour costs $12.00
1
2
hour costs $6.00
Answers
2 hours costs 6 dollars is the only statements that is true
--
simply add 3.00 to 3.00 because its 2 hours
or if it was half an hour simply add 1.50 because half of 3.00 is 1.50
ASAP. Pls help Pls help Pls help Pls help Pls help Pls help Pls help Pls help Pls help Pls help Pls help Pls help Pls help ASAP Pls help Pls help Pls help Pls help Pls help Pls help Pls help Pls help Pls help Pls help Pls help Pls help Pls help Pls help Pls help Pls help Pls help Pls help Pls help Pls help Pls help Pls help Pls help ASAP Pls help Pls help Pls help Pls help Pls help Pls help Pls help ASAP Pls help Pls help Pls help Pls help Pls help Pls help Pls help Pls help Pls help Pls help Pls help Pls help Pls help Pls help ASAP
Use the net to find the surface area of the square pyramid
Answers
Answer:
333 ft^2
Step-by-step explanation:
Surface area is the sum of all the areas of the shape.
First, the area of the square in the center is simply 9x9 = 81.
The area of a triangle is (bh)/2, where b = base, and h = height
In the triangle, 9 is the base, and 14 is the height:
(9x14)/2 = 63
There are four triangles of the same area:
63 x 4 = 252
252 + 81 = 333
Answer:
333ft^2
Step-by-step explanation: