Assuming you meant "x - 2(x - 2) * 3" as the expression, here is how you can simplify it:
x - 2(x - 2) * 3= x - 2(3x - 6) // Distribute the 2
= x - 6x + 12 // Distribute the -2
= -5x + 12 // Simplify
Therefore, the simplified expression is -5x + 12.
ANOTHER SCENARIOTo simplify the expression "x subtract 2 x subtract 2 multiply 3," we need to follow the order of operations, which is Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).
First, we need to simplify the multiplication by 3 inside the parentheses:
x - 2x - 2(3)Next, we can simplify the subtraction of 2x:
-x - 2(3)
Finally, we can simplify the multiplication of 2 and 3:
-x - 6
Therefore, the simplified expression is "-x - 6".
HELP I NEED THIS ANSWERED PORFAVOR
Answer:
3) x = 19
4) x = 4
Step-by-step explanation:
3) Co interior angles are supplementary.
6x - 30 + 96 = 180
Combine like terms,
6x + 66 = 180
Subtract 66 from both sides,
6x = 180 - 66
6x = 114
Divide both sides by 6,
x = 114 ÷ 6
[tex]\boxed{\bf x = 19}[/tex]
4) When two lines are intersected by a transversal, alternate exterior angles are equal.
7x + 5 = 33
Subtract 5 from both sides,
7x = 33- 5
7x = 28
Divide both sides by 7,
x = 28 ÷ 7
[tex]\boxed{\bf x = 4}[/tex]
The angles and their converse are:
a) ∠9 ≅ ∠22
Parallel lines: q and r.
Converse: if two lines are parallel, then their corresponding angles are congruent.
b) m∠8 + m∠13 = 180
Parallel lines: p and q.
Converse: if two lines are parallel, then their supplementary angles sum up to 180°
c) ∠1 ≅ ∠17
Parallel lines: p and r.
Converse: if two lines are parallel, then their corresponding angles are congruent.
d) ∠16 ≅ ∠20
Parallel lines: q and r.
Converse: if two lines are parallel, then their alternate angles are congruent.
e) m∠2 + m∠22 = 180
Parallel lines: p and r.
Converse: if two lines are parallel, then their supplementary angles sum up to 180°
f) ∠10 ≅ ∠13
No lines are parallel.
Converse: if two angles in a quadrilateral are adjacent then they are said to be congruent.
g) ∠5 ≅ ∠15
Parallel lines: p and q.
Converse: if two lines are parallel, then their opposite angles are congruent.
h) m∠2 + m∠16 = 180
Parallel lines: p and q.
Converse: if two lines are parallel, then their supplementary angles sum up to 180°
How to find the missing angles?a) ∠9 ≅ ∠22
The parallel lines for this to be true are lines q and r.
Converse is if two lines are parallel, then their corresponding angles are congruent.
b) m∠8 + m∠13 = 180
The parallel lines for this to be true are lines p and q.
Converse is if two lines are parallel, then their supplementary angles sum up to 180°
c) ∠1 ≅ ∠17
The parallel lines for this to be true are lines p and r.
Converse is if two lines are parallel, then their corresponding angles are congruent.
d) ∠16 ≅ ∠20
The parallel lines for this to be true are lines q and r.
Converse is if two lines are parallel, then their alternate angles are congruent.
e) m∠2 + m∠22 = 180
The parallel lines for this to be true are lines p and r.
Converse is if two lines are parallel, then their supplementary angles sum up to 180°
f) ∠10 ≅ ∠13
No lines are parallel.
Converse is if two angles in a quadrilateral are adjacent then they are said to be congruent.
g) ∠5 ≅ ∠15
The parallel lines for this to be true are lines p and q.
Converse is if two lines are parallel, then their opposite angles are congruent.
h) m∠2 + m∠16 = 180
The parallel lines for this to be true are lines p and q.
Converse is if two lines are parallel, then their supplementary angles sum up to 180°
3) The co interior angles are supplementary.
Thus:
6x - 30 + 96 = 180
6x + 66 = 180
Subtract 66 from both sides,
6x = 180 - 66
6x = 114
Divide both sides by 6,
x = 24
4) When two lines are intersected by a transversal, the alternate exterior angles are equal.
7x + 5 = 33
Subtract 5 from both sides,
7x = 33- 5
7x = 28
Divide both sides by 7,
x = 4
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Which of the following statements is NOT true? (graph in the picture)
The slope of the line of best fit is 4,744.
The y-intercept of the line of best fit is 9,994.
The slope of the line of best fit means that the value of the house increased
$4,744 each year.
The y-intercept of the line of best fit means that it cost $9,994 to build the house.
The statement that is NOT true is: "The y-intercept of the line of best fit means that it cost $9,994 to build the house."
Hi! Based on the given information and without a visual of the graph, I will provide a general answer using the terms provided. Please note that an accurate response may require additional information or context.
The slope of the line of best fit indicates the rate of change in the value of the house over time, and a positive slope signifies that the value has increased. However, the y-intercept of the line of best fit does not necessarily mean that it cost $9,994 to build the house.
The y-intercept represents the estimated value of the house when the independent variable (usually time) is zero, but it does not account for factors like construction costs, inflation, or changes in the real estate market.
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On a coordinate plane, trapezoid A B C D has points (negative 3, negative 2), (negative 1, 2), (3, 2), and (5, negative 2).
Figure ABCD is graphed on a coordinate plane.
ABCD is an isosceles trapezoid. What is the approximate perimeter of ABCD? Round to the nearest tenth.
16.5 units
18.9 units
20.9 units
24.0 units
16.5 I relllllllly hope this helps
3.9+3.4+4.5+3.1+3.9+4.1+3.6+3.9
Answer: 30.4
Step-by-step explanation:
Cubie Corporation has provided the following data concerning its only product: Selling price $ 87 per unit Current sales 13,000 units Break-even sales 11,700 units What is the margin of safety in dollars?
The margin of safety in dollars is $113100
The margin of safety is the value of sales or sales in units that are in excess of the break-even sales or units. The break-even point is where the firm is neither making a profit nor a loss and its total revenue is equal to its total expenses.
The margin of safety can be calculated by deducting the break-even sales value from the budgeted/actual sales value.
The margin of safety in dollars = budgeted/actual sales value - break-even sales value
Margin of safety in dollars = (87 * 13000) - (87 * 11700) = $113100
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Which of the following is a balance for a single $1856 deposit in an account with an APR of 2.42% that compounds interest quarterly and is invested for six
Answer:
Using the formula for compound interest: A = P(1 + r/n)^(nt) where: A = the balance after the investment period P = the principal amount (the initial deposit) r = the annual interest rate (as a decimal) n = the number of times the interest is compounded per year t = the time the money is invested (in years) We can plug in the values given in the problem: P = $1856 r = 0.0242 (2.42% expressed as a decimal) n = 4 (quarterly compounding) t = 6/12 (6 months expressed as a fraction of a year) A = $1856(1 + 0.0242/4)^(4 * 6/12) A = $1856(1.00605)^2 A = $1931.
use the points (6,56) and (12,26) from the following data set to determine the point-slope form of an equation that represents the data set. x 5 6 7 8 10 11 12 12 14 y 58 56 53 46 37 33 26 23 13 answer
The point-slope form of the equation that represents the data set is y - 56 = -5(x - 6), or y = -5x + 86.
EquationsWe are given two points: (6,56) and (12,26) from the data set. We can use the point-slope form of a linear equation to find an equation that represents the data set.
y - y1 = m(x - x1)
where (x1, y1) is a point on the line and m is the slope of the line.
To find the slope, we use the two given points:
m = (y2 - y1) / (x2 - x1)
m = (26 - 56) / (12 - 6)
m = -30 / 6
m = -5
Now that we have the slope, we can use one of the given points, say (6, 56), to write the point-slope equation:
y - y1 = m(x - x1)
y - 56 = -5(x - 6)
y - 56 = -5x + 30
y = -5x + 86
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What will be the new position of the given point (-4,-3) after rotating 180° about the origin?
Answer:
(4,3)
Step-by-step explanation:
The rule (x,y) → (-x, -y)
Helping in the name of Jesus.
64. A polyhedron has 4 vertices and 4 faces. How
many edges does the polyhedron have?
(F) 4
(G) 5
(H) 6
(J) 8
(K) 16
Answer:
(H)
Step-by-step explanation:
Recall that [tex]V-E+F=2[/tex].
Setting [tex]V=F=4[/tex] yields [tex]E=6[\tex].
URGENT PLEASEEE!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
Neil is creating a computer game in which bubbles represented by circles collide, merge, and separate in different ways. A bubble with a radius of 8 centimeters separates into small bubbles, each of which has a radius of 2 centimeters. The area of the large bubble is equal to the sum of the areas of the small bubbles. How many small bubbles are there?
There will be 16 small circular bubbles.
What is circle?A circle is a shape formed by all points in a plane that are at a particular distance from the centre. It is the curve sketched out by a point moving in a plane so that its distance from a given point remains constant. The radius is the distance between any two points on a circle and the centre.
The area of a circle is given by the formula A = πr², where A is the area and r is the radius.
The area of the large bubble is A = π(8cm)² = 64π cm².
The area of a small bubble is A = π(2cm)² = 4π cm².
Let's assume that the number of small bubbles is n. Then, the total area of the small bubbles is n times the area of a single small bubble:
n(4π) = 4nπ cm²
According to the problem, the area of the large bubble is equal to the sum of the areas of the small bubbles:
64π = 4nπ
Dividing both sides by 4π, we get:
n = 16
Therefore, there are 16 small bubbles.
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-2/3w=12 solve for w
Two functions are given below: f(x) and h(x). State the axis of symmetry for each function and explain how to find it.
The axis of symmetry for f(x) is x = 1 and h(x) is x=3.
What is axis of symmetry?The axis of symmetry is an imaginary straight line that divides a shape into two identical parts, thereby creating one part as the mirror image of the other part. When folded along the axis of the symmetry, the two parts get superimposed. The straight line is called the line of symmetry/the mirror line. This line can be vertical, horizontal, or slanting.
Equation:Assuming that f(x) and h(x) are quadratic functions, here's how to find the axis of symmetry for each function:
f(x) = 2x² - 4x + 1
To find the axis of symmetry for f(x), we first identify the coefficients a, b, and c:
a = 2, b = -4, c = 1
Then, we use the formula:
x = -b / (2a) = -(-4) / (2 * 2) = 1
Therefore, the axis of symmetry for f(x) is x = 1. This means that the graph of f(x) is symmetric with respect to the vertical line x = 1.
h(x) = -x² + 6x - 5
To find the axis of symmetry for h(x), we first identify the coefficients a, b, and c:
a = -1, b = 6, c = -5
Then, we use the formula:
x = -b / (2a) = -6 / (2 * -1) = 3
Therefore, the axis of symmetry for h(x) is x = 3. This means that the graph of h(x) is symmetric with respect to the vertical line x = 3.
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The complete question would be:
Two functions are given below: f(x)=2x² - 4x + 1 and h(x)=-x² + 6x - 5. State the axis of symmetry for each function and explain how to find it.
help ASAP
Using a standard deck of cards, a gamer drew one card and recorded its value. They continued this for a total of 100 draws. The table shows the frequency of each card drawn.
Card A 2 3 4 5 6 7 8 9 10 J Q K
Frequency 4 7 5 6 7 6 8 10 7 10 8 12 10
Based on the table, what is the experimental probability that the card selected was a K or 6?
one over 26
four over 25
one fourth
8 over 13
The experimental probability that the card selected was a K or 6 is [tex]3/20[/tex].
The answer is "three over twenty."
What is probability?The frequency of K and 6 in the table are 8 and 7 respectively. Therefore, the total number of times a K or 6 was drawn is [tex]8 + 7 = 15[/tex].
The total number of draws was 100, so the probability of drawing a K or 6 on any one draw is:
Probability of K or 6 =[tex](Frequency of K + Frequency of 6) / Total number of draws[/tex]
[tex]= (8 + 7) / 100\\= 15 / 100\\= 3 / 20[/tex]
So the experimental probability that the card selected was a K or 6 is [tex]3/20[/tex].
Therefore, the answer is "three over twenty."
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Make a new triangle
B
15
Ch
17
8
The Finale! All trig ratios for a right triangles
Complete the table below using what you know about
trigonometric ratios for right triangles.
LA
Write your ratios as fractions. A message will appear
when you are correct.
angle
LB
opp
adj
hyp
sin
COS
8
17
15
17
tan
15
8
15
8
Angle A is the right angle, so the trigonometric ratio we get is sin(LA) = 8/17, cos(LA) = 15/17, tan(LA) = 8/15, sin(LB) = 15/17, cos(LB) = 8/17, tan(LB) = 15/8
What is the trigonometric ratio?Sine (sin), cosine (cos), tangent (tan), cotangent (cot), cosecant (cosec), and secant are the six trigonometric ratios. (sec). A branch of mathematics called trigonometry in geometry works with the sides and angles of a right-angled triangle. Trig ratios are therefore assessed in relation to sides and angles.
The right triangle with sides that are 8, 15, and 17 in length needs to be labeled as follows in order to use trigonometry ratios.
B
/|
17/ |
/ |
/___|__
A C 15
where the right angle A is. The chart can then be filled out as follows:
angle opp adj hyp sin cos tan
LA 8 15 17 8/17 15/17 8/15
LB 15 8 17 15/17 8/17 15/8
The tangent is opposite/adjacent, the sine is adjacent/hypotenuse, and the cosine is opposite/hypotenuse.
The trigonometry ratios for the right triangle with sides that are 8, 15, and 17 in length are as follows.
sin(LA) = 8/17
cos(LA) = 15/17
tan(LA) = 8/15
sin(LB) = 15/17
cos(LB) = 8/17
tan(LB) = 15/8
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d = 8 1/2 in., r = underline ? , underline C =?
Given: D = 8 1/2 in.
We know that the diameter of a circle is twice the radius. Therefore, we can find the radius by dividing the diameter by 2:
radius (r) = D/2 = 8 1/2 / 2 = 4 1/4 in.
To find the circumference (C) of the circle, we can use the formula:
C = 2πr
where π (pi) is a constant approximately equal to 3.14.
Substituting the value of r, we get:
C = 2π(4 1/4) = 2π(17/4) = 8.5π
Therefore, the radius is 4 1/4 in. and the circumference is 8.5π in.
Find the greatest common factor of 3, 15, and 35.
the gcf of 3, 15, and 35 is 1.
Answer: 1
Step-by-step explanation:
GCF means greatest common factor.
And the greatest common factor of 3, 15 and 35 would be 1 since
no other number is common between them.
3 * 1 = 3
15 * 1 = 15
35 * 1 =35
The ratio of A to B is 1 : 2 and B to C is 3 : 4. If C has the value of 32, what is the sum of the three numbers?
Answer:
A + B + C = 68
Step-by-step explanation:
A : B = 1 : 2
B : C = 3 : 4
Find the LCM of 2 and 3. LCM = 6
Multiply A : B by 3 and B : C by 2
[tex]\dfrac{A}{B}=\dfrac{1}{2}=\dfrac{1*3}{2*3}=\dfrac{3}{6}\\\\\\\dfrac{B}{C }=\dfrac{3}{4}=\dfrac{3*2}{4*2}=\dfrac{6}{8}[/tex]
A : B = 3 : 6 & B : C = 6 :8
A : B : C = 3 : 6 : 8
A = 3x
B = 6x
C = 8x
It is given that C is 32.
8x = 32
Divide both sides by 8
x = 32÷ 8
x = 4
A = 3*4 = 12
B = 6 * 4 = 24
Sum of three numbers = 12 + 24 + 32
= 68
0.85 divided by 3 2/5 divided by 3/4
Answer: 1/3
Step-by-step explanation: To do complex equations like this try using a calculator
How would you solve this?
Answer:
average rate of change = [tex]\frac{1}{2}[/tex]
Step-by-step explanation:
the average rate of change of h(x) in the interval a ≤ x ≤ b , is
[tex]\frac{h(b)-h(a)}{b-a}[/tex]
here the interval is - 2 ≤ x ≤ 2 , then
h(b) = f(2) = [tex]\frac{1}{8}[/tex] (2)³ - 2² = [tex]\frac{1}{8}[/tex] (8) - 4 = 1 - 4 = - 3
h(a) = h(- 2) = [tex]\frac{1}{8}[/tex] (- 2)³ - (- 2)² = [tex]\frac{1}{8}[/tex] (- 8) - 4 = - 1 - 4 = - 5
then average rate of change
= [tex]\frac{-3-(-5)}{2-(-2)}[/tex]
= [tex]\frac{-3+5}{2+2}[/tex]
= [tex]\frac{2}{4}[/tex]
= [tex]\frac{1}{2}[/tex]
significant figure of 9,876 ,540,000
Answer:
The number you provided is **9,876 ,540,000**. The significant figures of this number are **10**¹.
The trailing zeros in this number are not significant, so they do not count toward the total number of significant figures
−2x + 1 ≥ 5. show work inequality
Answer:
x≥ -2
Step-by-step explanation:
−2x + 1 ≥ 5
-1 -1
-2x ≥ 4
x≥ -2
If the prisms below are similar, what is the ratio of the height of the smaller prism to the height of the larger prism? V = 15 cm3 V = 405 cm3
The ratio of the height of the smaller prism to the height of the larger prism is 1:sqrt(27).
what is prism?
A prism is a geometric solid that has two parallel and congruent polygonal bases, and the sides connecting the bases are parallelograms. These parallelograms are typically rectangles, but they can also be squares or rhombuses. A prism can be named based on the shape of its base, such as a rectangular prism, triangular prism, hexagonal prism, etc.
Since the prisms are similar, their corresponding sides are proportional. In particular, the ratio of the heights of the prisms is the same as the ratio of their volumes.
Let the height of the smaller prism be h1 and the height of the larger prism be h2. Let the volume of the smaller prism be V1 = 15 cm^3 and the volume of the larger prism be V2 = 405 cm^3. Then:
V1/V2 = (h1^2 * 5) / (h2^2 * 5)
V1/V2 = h1^2 / h2^2
sqrt(V1/V2) = h1 / h2
Substituting the given values, we get:
sqrt(15/405) = h1 / h2
sqrt(1/27) = h1 / h2
1/sqrt(27) = h1 / h2
Therefore, the ratio of the height of the smaller prism to the height of the larger prism is 1:sqrt(27), or approximately 1:5.2.
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A sample of students is taken from the school's A honor roll. The school estimates that there are actually 360 students on the A honor roll. Using this sample from the table, how many students on the A honor roll are 8th graders?
280 8th graders
172 8th graders
114 8th graders
126 8th graders
The estimated number of students on the A honor roll who are 8th graders is 126.
What is Algebra?Algebra is a branch of mathematics that deals with symbols and the rules for manipulating these symbols. It involves using letters and other symbols to represent numbers and quantities in equations and formulas.
The main goal of algebra is to find the value of an unknown quantity, called a variable, by using known quantities and mathematical operations such as addition, subtraction, multiplication, division, and exponentiation. Algebraic expressions can be solved using various techniques, including simplification, factoring, and solving equations.
The total number of students on the A honor roll is given by adding the number of students in each grade:
Total = 15 (6th grade) + 11 (7th grade) + 14 (8th grade) = 40
To find the proportion of students who are in 8th grade, we can divide the number of 8th graders by the total number of students:
Proportion of 8th graders = 14/40 = 0.35
To estimate the number of students on the A honor roll who are 8th graders, we can multiply the proportion of 8th graders by the total number of students on the A honor roll:
Estimated number of 8th graders = 0.35 x 360 = 126
Therefore, the estimated number of students on the A honor roll who are 8th graders is 126.
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Cam anyone answer these 4 questions?
Answer:
The first one is correct
Step-by-step explanation:
ITs all good
What happens to the surface area of a rectangular prism if you double one dimension? Let's create several examples and see if a pattern emerges.
Let's take a look at Karen's post.
My rectangular prism has sides of 8 units, 6 units, and 5 units. To find the total surface area of the prism, I have to find the area of each face of the prism. There are two faces that have an area of 8(5) = 40 units2, two that have an area of 6(5) = 30 units2, and two that have an area of 8(6) = 48 units2. Adding all the areas I get the surface area is 236 units2.
If the smallest side is doubled the dimensions would be 8 units, 6 units, and 10 units. There are two faces that have an area of 8(10) = 80 units2, two that have an area of 6(10) = 60 units2, and two that have an area of 8(6) = 48 units2. Adding all the areas I get the surface area is 376 units2.
The surface area increased by 140 units2 or a factor of about 1.6.
Now it's your turn.
Come up with a set of dimensions for a rectangular prism, and calculate its surface area.
Double one of the dimensions of your prism. List the new set of dimensions, and find the new surface area.
Describe how doubling one of the dimensions affects the surface area.
Look at your classmates' posts. Add a response to one of the posts stating any patterns you noticed in the calculations of all your classmates, and the effect of doubling on dimension on the surface area.
Use the discussion rubric to see how you will be graded.
Answer:
All you need to do now is plug the numbers in and use the right units.
Step-by-step explanation:
When 18 liters of oil were poured out, the oil level went down by 90 %. How much oil was in the tank at first?
What is the volume of the solid figure?
Answer:
[tex]625 \: {in}^{3} [/tex]
Step-by-step explanation:
This figure is formed from a cube and a rectangular prism
First, we can find the volume of the cube:
(l (side length) = 5 in)
[tex]v(cube) = {l}^{3 } = {5}^{3} = 125 \: {in}^{3} [/tex]
Now, let's find the volume of the rectangular prism:
(h = 5 in; a (base area) = 20 × 5 = 100 in^2)
[tex]v(prism) = a(base) \times h[/tex]
[tex]v(prism) = 100 \times 5 = 500 \: {in}^{3} [/tex]
In order to find the total volume of this figure, we have to add these two volumes together:
[tex]v(total) = v(cube) + v(prism)[/tex]
[tex]v(total) = 125 + 500 = 625 \: {in}^{3} [/tex]
what is the answer to this question?
dy/dx=?
[tex] \:\:\:\:\: \:\:\:\:\:\:\star\longrightarrow \sf y = x^{x}{}^{²}\\[/tex]
Taking the logarithm on both sides -
[tex] \:\:\:\:\: \:\:\:\:\:\:\longrightarrow \sf log y = log x^{x}{}^{²}\\[/tex]
[tex] \:\:\:\:\:\:\:\:\:\:\:\longrightarrow \sf log y = x^2 log x\\[/tex]
[tex]\:\:\: \boxed{\sf\pink{\:\:\: loga^b = blog a }}\\[/tex]
Differentiating with respect to x-
[tex] \:\:\:\:\:\:\:\:\:\:\:\longrightarrow \sf \dfrac{d}{dx} logy = \dfrac{d}{dx} x^2 log x \\[/tex]
[tex] \:\:\:\:\: \:\:\:\:\:\:\longrightarrow \sf \dfrac{1}{y} \times \dfrac{dy}{dx} = x^2 \dfrac{d}{dx} log x + logx \dfrac{d}{dx} x^2\\[/tex]
[tex] \:\:\:\:\boxed{\sf\pink{\dfrac{d}{dx} logx = \dfrac{1}{x}}} \\[/tex]
[tex] \:\:\:\:\boxed{\sf\pink{\sf\dfrac{d}{dx}\bigg[f(x)\:g(x)\bigg] = f(x) \dfrac{d}{dx} g(x) + g(x) \dfrac{d}{dx} f(x)}}\\[/tex]
[tex] \:\:\:\:\: \:\:\:\:\:\:\longrightarrow \sf \dfrac{d}{dx} = y \bigg[ x^2 \times \dfrac{1}{x} + logx \times 2x \bigg]\\[/tex]
[tex] \:\:\:\:\:\:\:\:\:\:\:\longrightarrow \sf \dfrac{dy}{dx} = y \bigg[ \cancel{x}\: x \times \dfrac{1}{\cancel{x}} + 2x\:logx \bigg]\\[/tex]
[tex] \:\:\:\:\:\:\:\:\:\:\:\longrightarrow \sf \underline{\dfrac{dy}{dx} = y \bigg[ x + 2x\:logx \bigg]}\\[/tex]
[tex] \:\:\:\:\:\:\:\:\:\:\:\longrightarrow \sf \underline{\dfrac{dy}{dx} = \boxed{\sf x^{x}{}^{²}\bigg[ x + 2x\:logx \bigg]}}\\[/tex]
There are three possible cases (or scenarios) for how many solutions that an absolute value equation could have. How many solutions are there for each case? Why are their differences in the number of solutions? Give a mathematical example in your explanation.
PLS HELP ASAP
Consider the equation |x-4| = 5. This equation has two solutions since the distance of x from 4 on the number line can be 5 units in either direction, giving x = 9 or x = -1.
What is algebra?
Algebra is a branch of mathematics that deals with mathematical operations and symbols used to represent numbers and quantities in equations and formulas.
An absolute value equation is of the form |x| = a, where "a" is a positive number. The absolute value of a number is always non-negative, so the equation |x| = a has two possible solutions, x = a and x = -a.
However, there are three possible cases for how many solutions an absolute value equation could have:
One solution: This occurs when a=0. The only solution is x=0 because |0|=0.
Two solutions: This occurs when a > 0. The two possible solutions are x = a and x = -a, since |a| = a and |-a| = a. For example, the equation |x-3| = 2 has two solutions: x-3 = 2 or x-3 = -2, which gives x = 5 or x = 1.
No solutions: This occurs when a < 0. Since the absolute value of a number is always non-negative, an absolute value equation with a negative number on the right-hand side has no solutions. For example, the equation |x-3| = -2 has no solutions since -2 is negative.
The difference in the number of solutions is due to the nature of absolute values. The absolute value of a number represents the distance of the number from zero on the number line, so an absolute value equation can have two solutions when the distance is equal to a positive number, one solution when the distance is equal to zero, and no solution when the distance is less than zero.
For example, consider the equation |x-4| = 5. This equation has two solutions since the distance of x from 4 on the number line can be 5 units in either direction, giving x = 9 or x = -1.
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Answer: Absolute value equations are important in situations where values cannot be negative, like measuring distance. For example, if you forget which floor your friend lives on and he tells you he's on the fourth floor, and you say you're two floors away, you could be on either the second or sixth floor. Absolute value inequalities are important in determining margins of error or tolerance, especially in manufacturing.
Step-by-step explanation:
When solving an absolute value equation, three possible scenarios could occur. The first scenario is when the absolute value expression equals a positive number. In this case, there will be two solutions, one positive and one negative. The second scenario is when the absolute value expression equals zero. In this case, there will be only one solution, which is zero. The third and final scenario is when the absolute value expression equals a negative number. In this case, there are no solutions, as the absolute value of any number is always non-negative.
The reason there are differences in the number of solutions for each case is due to the nature of absolute value. Absolute value always returns a non-negative value, regardless of the sign of the number inside the absolute value expression. Therefore, when the absolute value expression is positive, there are two possible solutions, one positive and one negative. When the expression equals zero, there is only one solution, which is zero. And when the expression is negative, there are no solutions, as there cannot be a negative absolute value.
Select the correct answer from each drop-down menu.
Brian and Leo are flying to their grandmother's house on an airplane where 50 out of the 150 seats are window seats and passengers are randomly
assigned seats on the plane for each flight.
which is
the probability that Brian is assigned a window seat on the flight to his grandmother's house and the flight home from his
The probability that both Brian and Leo are both assigned window seats on the way to their grandmother's house is
grandmother's house.
The probabilities of Brian and leo being assigned a window seat on each flight are 1/9
what is the probability that Brian is assigned?The probability that Brian is assigned a window seat on the flight to his grandmother's house is:
50/150 = 1/3
The probability that LEO is assigned a window seat on the flight home from his grandmother's house is also:
50 window seats/ 150 seats = 1/3
The probability that both Brian and Leo are both assigned window seats on the way to their grandmother's house is:
50 window seats/150 seats * 49 window seats/ 149 seats = 1/9
Take note that Brian and Leo's chances of getting a window seat on each flight are independent, so we can simply multiply the odds to determine the likelihood of both happening.
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