Which linear inequality is represented by the graph?
y < 2/3x + 3
y >3/2 x + 3
y >2/3 x + 3
y < 3/2x + 3
Answers
The linear equality represented by the graphis [tex]\boxed{{\mathbf{y > }}\frac{{\mathbf{2}}}{{\mathbf{3}}}{\mathbf{x + 3}}}[/tex] and it matches with [tex]\boxed{{\mathbf{OPTION C}}}[/tex].
Further explanation:
From given graph, theline passes through points [tex]\left({3,5}\right)[/tex] and [tex]\left({ - 3,1}\right)[/tex] as shown below in Figure 1.
The slope of a line passes through points [tex]\left({{x_1},{y_1}}\right)[/tex] and [tex]\left({{x_2},{y_2}}\right)[/tex] is calculated as follows:
[tex]m = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}{\text{}}[/tex] ......(1)
Here, the slope of a line is denoted as and points are [tex]\left({{x_1},{y_1}}\right)[/tex] and [tex]\left({{x_2},{y_2}}\right)[/tex].
Substitute [tex]3[/tex] for [tex]{x_1}[/tex] , [tex]5[/tex] for [tex]{y_1}[/tex] , [tex]-3[/tex] for [tex]{x_2}[/tex] and [tex]1[/tex] for [tex]{y_2}[/tex] in equation (1) to obtain the slope of a line that passes through points [tex]\left( {3,5}\right)[/tex] and [tex]\left({ - 3,1}\right)[/tex].
[tex]\begin{aligned}m&=\frac{{1 - 5}}{{ - 3 - 3}}\\&=\frac{{ - 4}}{{ - 6}}\\&=\frac{2}{3}\\\end{aligned}[/tex]
Therefore, the slope is [tex]\frac{2}{3}[/tex].
The point-slope form of the equation of a line with slope passes through point is represented as follows:
[tex]y - {y_1} = m\left({x - {x_1}}\right){\text{}}[/tex] ......(2)
Substitute [tex]3[/tex] for [tex]{x_1}[/tex] , [tex]5[/tex] for [tex]{y_1}[/tex] and [tex]\frac{2}{3}[/tex] for [tex]m[/tex] in equation (2) to obtain the equation of line.
[tex]\begin{aligned}y - 5 = \frac{2}{3}\left({x - 3} \right)\\3\left( {y - 5} \right) = 2x - 6\\3y - 15 = 2x - 6\\3y = 2x - 6 + 15\\\end{aligned}[/tex]
Further simplify the above equation.
[tex]\begin{aligned}3y=2x + 9\\y=\frac{2}{3}x+\frac{9}{3}\\y=\frac{2}{3}x + 3\\\end{aligned}[/tex]
Therefore, the value of [tex]y[/tex] is [tex]\dfrac{2}{3}x + 3[/tex].
Since the shaded part in Figure 1 is above the equation of line [tex]y = \dfrac{2}{3}x + 3[/tex], therefore, greater than sign is used instead of is equal to
Therefore, the inequality is [tex]y > \dfrac{2}{3}x + 3[/tex].
Now, the four options are given below.
[tex]\begin{aligned}{\text{OPTION A}}\to y < \frac{2}{3}x + 3 \hfill \\{\text{OPTION B}} \to y > \frac{3}{2}x + 3 \hfill \\{\text{OPTION C}} \to y > \frac{2}{3}x + 3 \hfill \\{\text{OPTION D}}\to y < \frac{3}{2}x + 3 \hfill\\\end{aligned}[/tex]
Since OPTION C matches the obtained equation that is [tex]y > \dfrac{2}{3}x + 3[/tex].
Thus, the linear equality represented by the graph is [tex]\boxed{{\mathbf{y > }}\frac{{\mathbf{2}}}{{\mathbf{3}}}{\mathbf{x + 3}}}[/tex]and it matches with [tex]\boxed{{\mathbf{OPTION C}}}[/tex]
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Answer Details:
Grade: Junior High School
Subject: Mathematics
Chapter: Coordinate Geometry
Keywords: Coordinate Geometry, linear equation, system of linear equations in two variables, variables, mathematics,equation of line, line, passes through point, inequality
Related Questions
The average (arithmetic mean) of 5,10,15 and x is 20..What is the x ?
Answers
[tex]20=\frac{5+10+15+x}{4}[/tex]
[tex]80=30+x[/tex]
[tex]x=50[/tex]
Answer:
x = 50 .
Step-by-step explanation:
Given : Mean of 5,10,15 and x is 20.
To find : What is the x.
Solution : We have given Mean of 5, 10 , 15 and x is 20.
Mean = 20 .
Mean = [tex]\frac{sum\ of \ data}{Number\ of \ data}[/tex].
Mean = [tex]\frac{5+10+15 +x}{4}[/tex].
20 = [tex]\frac{30 +x}{4}[/tex].
On multiplying both sides by 4
20 * 4 = 30 + x .
80 = 30 + x .
On subtracting both sides by 30.
50 = x
Therefore, x = 50 .
What is the next term for the given arithmetic sequence 0,2,4,6
Answers
It's increasing by 2's
April works as a courier delivering and picking up packages for a shipping company. Three of the communities on her route form an isosceles right triangle as shown in the figure. There are approximately 48 miles between Old Mill and Clarksville and between Clarksville and Hamilton. Approximately how many miles are there between Old Mill and Hamilton? Round your answer to the nearest mile if necessary.
Answers
There will be two angles, equal to 45°, and a 90° angle as shown in the figure below.
The distance from Old Mill is the same as the distance from Clarksville to Hamilton, equal to 48 miles.
Let x = distance between Old Mill and Hamilton.
By definition,
cos 45° = 48/x
or
x = 48/cos 45° = 67.882 mi
Answer: 68 miles (nearest mile)
Solve the equation. 20x-7=5+16x
Answers
Add 7 to both sides
20x= 12+16x
Subtract 16x from both sides
4x=12
Divide both sides by 4
x=3
Final answer: x=3
20x - 7 = 16x + 5 (This is the same equation, I have just rearranged it to my preference)
First, we need to get x on one side, then isolate x.
We have x on both sides, so we need to get them on one side.
20x - 7 = 16x + 5
Subtract 16x from both sides.
20x - 16x = 4x
16x - 16x = 0
4x - 7 = 5
Now we need to isolate x.
Since 4x is being subtracted by 7, we need to add 7 to cancel it out.
Add 7 to both sides.
-7 + 7 = 0
5 + 7 = 12
4x = 12
Now that we have isolated x, we need to solve for x.
Since x is being multiplied by 4, we need to cancel it out by performing the opposite order of operations.
The opposite of multiplying is dividing.
Divide both sides by 4 to solve for x.
4x / 4 = x
12 / 4 = 3
We are left with our solution:
x = 3
I hope this helps!
Define a function f : R → R by the formula
f(x) = 2x − 8.
Prove that f is one-to-one.
Let a, b is in R, and suppose that
f(a) = f(b).
Then
2a − 8 =
Answers
Now, we have to think that the parent function is f(x) = 2x - 8 and f (a) = f (b). This means the value of the function f(a) is if a is substituted to x:
f (a) = 2a - 8, same goes for f (b):
f (b) = 2b - 8.
Since we have a condition that f(a) = f (b), then,
2a - 8 = 2b - 8.
If we simplify further,
2a = 2b - 8 + 8
( 2a = 2b ) / 2
the equation becomes:
a = b; this proves that f is a one-to-one function.
Elevator 1 in a building moved from ground position to a final position of +17 feet. Elevator 2 in the same building moved from ground to a final position of −22 feet. Which statement best describes the final positions of these two elevators?
Answers
Answer:Elevator is +17 above ground level while -22 is below
Step-by-step explanation:
Since it is + and - i know +is positive that mean up and - is negative which mean down so -22 is below while +16 is above
Write the number in scientific notation.0.00000184
Answers
= 1.84 x 10^-6
hope it helps
If T is the point at a distance 3pi/5 from P(1, 0) on the unit circle with its center at (0, 0), in what quadrant does T lie?
Answers
We know that 180 degrees on the unit circle is pi, or 5pi/5, which is the same as pi.
So 90 degrees is half of 180 and therefore half of 5pi/5 which is 2.5pi/5.
Since 2.5pi/5 is at the 90 degree mark we know that 3pi/5 has to be past this so it will be in Quadrant II.
I hope this helps.
Final answer:
The point T lies in the second quadrant of the unit circle when it is at an angular distance of 3π/5 from P(1, 0), because this angle falls between π/2 and π radians.
Explanation:
If T is the point at a distance 3π/5 from point P(1, 0) on the unit circle with its center at (0, 0), to determine in what quadrant T lies, we must consider where this angle places T on the unit circle.
Recall that a point on the unit circle has coordinates (x, y) = (cos θ, sin θ), where θ is the angle measured counterclockwise from the x-axis. An angle of 3π/5 radians is more than π/2 radians but fewer than π radians. Therefore, it falls in the second quadrant of the unit circle, where x-coordinates are negative and y-coordinates are positive.
Kayla has a tshirt store. she sold three times as many white tshirts as blue tshirts. she sold a total of 48 tshirts. How many white tshirts did kayla sell
Answers
1. x
2. 3x
x + 3x = 48
Now solve it:
x + 3x = 48
4x= 48
x=12
Now plug in x
1. x = 12
2. 3x = 36
So Kayla sold 12 blue t-shirts and 36 white t-shirts.
For which value of x should the expression be further simplified
√14x
56
11
5
Or 39
Answers
because it could be simplified to 28 where as the others would become decimals
The sum of 5 consecutive integers is 505
Answers
505/5 = 101
so 101 is the middle number, now add the 2 below it and the 2 above it
99 + 100 + 101 + 102 +103 = 505
Why is expected income a large factor in choosing a retirement plan
Answers
Expected income is crucial in choosing a retirement plan because it influences permanent income for savings, the adequacy of funds against inflation, and financial security post-retirement.
Choosing a retirement plan is a vital financial decision; expected income plays a large role because it directly affects a person's ability to save and sustains their lifestyle after they retire. The idea of permanent income, which is the average annual income an individual expects to receive for the rest of their lives, influences how much they allocate to retirement savings. It's crucial for asset accumulation and ensuring financial security.
Retirement funds sourced from plans like private company pensions or defined benefits can be fixed and may not keep up with inflation, leading to a potential loss of purchasing power over time. Therefore, understanding one's expected income helps in selecting retirement plans that can adjust and possibly increase over time to hedge against inflation. Additionally, individuals must consider the impact of future income changes on their current saving and consumption patterns.
Behavioral economics factors in human psychology and behavior, suggesting that individuals may not always make rational financial decisions due to a lack of self-control or understanding of long-term benefits, influencing retirement plan choices. Thus, expected income is a significant consideration for individuals when choosing a retirement plan, as it determines their ability to save enough to maintain their desired lifestyle in retirement.
How to find the distance between a point and a line
Answers
Final answer:
To find the distance between a point and a line, you need to find the equation of the line, find the coordinates of the point, substitute the coordinates into the equation, and calculate the perpendicular distance.
Explanation:
To find the distance between a point and a line, you need to use the formula for the distance from a point to a line. Here are the steps:
First, find the equation of the line.
Next, find the coordinates of the given point.
Substitute the coordinates of the point into the equation of the line to find the perpendicular distance from the point to the line.
The distance between the point and the line is the perpendicular distance found in step 3.
For example, let's say the equation of the line is y = 2x + 3, and the coordinates of the point are (4,5). To find the distance between the point and the line:
The equation of the line is y = 2x + 3.
The coordinates of the point are (4,5).
Substituting the coordinates (4,5) into the equation y = 2x + 3 gives 5 = 2(4) + 3, which simplifies to 5 = 11.
The perpendicular distance from the point to the line is |5 - 11|, which is equal to 6.
Therefore, the distance between the point (4,5) and the line y = 2x + 3 is 6 units.
how else can the ratio 14 to 1 be written
Answers
:)
Which of the following inequalities represents a number n that is less than or equal to 3 but greater than –2?
Answers
That's the inequality
5+1x10 is the answer 60 or 15? there are no parenthesis***
Answers
1x10= 10
10+5= 15
So you would do 1x10=10 and then add 5+10=15.
An acrobat of mass m slides down a vertical rope of height h. For the first three quarter of her descent she grips the rope with her hands and legs so as to produce a fictional force equal to five-ninth of her weight.she then tighten her grip so that she comes to rest at the bottom of the rope. Sketch a (t .v) graph to illustrate her descent
Answers
The acrobat's descent on a time versus velocity graph would show an initial linear increase in velocity due to gravity minus frictional force, followed by a sharp negative slope as she comes to a stop at the bottom.
The acrobat's descent can be analyzed by plotting a time (t) versus velocity (v) graph. Initially, the acrobat experiences a frictional force equal to five-ninths of her weight, which creates constant deceleration. For the first three quarters of the descent, her velocity will increase linearly (assuming constant deceleration) until she tightens her grip, which will cause a sharp decrease in velocity until she comes to a complete stop at the bottom. The graph would show a slope that decreases in magnitude over time and then drops to zero when the acrobat comes to rest.
Example Graph Description:The graph starts at the origin (0,0), indicating that the descent starts from rest.As time progresses, the velocity increases linearly due to the effect of gravity partially opposed by frictional force.Before reaching the bottom, the acrobat applies enough force to stop, leading to a sharp negative slope on the graph that touches the t-axis, indicating a stop.What value of x is in the solution set of 2x – 3 > 11 – 5x?
Answers
2x - 3 > 11 - 5x
2x > 14 - 5x <-- Add 3 to both sides
7x > 14 <-- Add 5x to both sides
x > 2 <-- Divide both sides 7
In order to be in the solution, x has to be greater than 2.
In interval notation: (2, ∞)
In set builder notation: {x | x ∈ R, x > 2}
a pair of jeans cost 75% as much as a pair of corduroy pants what is the cost of each if the total cost for the jeans and courds is $42
Answers
j= jeans
c= corduroy
jeans = 0.75C
0.75 +c = 42
1.75c = 42
c = 42/1.75 = 24
jeans cost 24*0.75 = 18
corduroys cost $24
Jeans cost $18
Answer with Step-by-step explanation:
Let j denote the cost of jeans
and c denotes the cost of corduroy
A pair of jeans cost 75% as much as a pair of corduroy pants.
i.e. j = 0.75c
The total cost for the jeans and corduroy is $42
i.e. j+c=42
⇒ 0.75c+c=42
⇒ 1.75c=42
Dividing both sides by 1.75,we get
c=24
Putting value of c in j= 0.75c gives
j=18
Hence, Cost of jeans=$18
and cost of corduroy=$24
Solve −2x2 +3x − 9 = 0
Answers
The quadratic is not factorable so quadratic formula must be used.
x = (-b + - √(b^2 - 4ac))/2a
a = -2, b = 3, c = -9
x = (-3 + - √(9 - 4*-2*-9))/-4
x = (-3 + - √(-63))/-4
x = -3 + - 3i√7
--------------
-4
x = 3 + 3i√7 x = 3 - 3i√7
------------ ------------
4 4
To solve the equation -2x^2 + 3x - 9 = 0, we can use the quadratic formula. This equation does not have any real solutions, but it has complex solutions involving imaginary numbers.
Explanation:To solve the equation -2x^2 + 3x - 9 = 0, we can use the quadratic formula. Substituting the values a = -2, b = 3, and c = -9 into the formula, we can then simplify to find the solutions for x.
The quadratic formula is given by:
x = (-b ± sqrt(b^2 - 4ac)) / (2a)
Substituting the values, we have:
x = (-3 ± sqrt(3^2 - 4(-2)(-9))) / (2(-2))
Simplifying further, we get:
x = (-3 ± sqrt(9 - 72)) / -4
x = (-3 ± sqrt(-63)) / -4
Since the square root of a negative number is not a real number, this equation does not have any real solutions. The equation has complex solutions, which involve the use of imaginary numbers.
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X^2 + 4x - 21 Factor into two binomials
PLZ I NEED THE ANSWER ;c
Answers
Proof:
7*-3=-21
7-3=4
hope this helps
What does the symbol "kg" represent in the metric system?
Answers
What does the symbol "kg" represent in the metric system?
"kilogram"
Hoped This Helped, InsomniaVoid
Your Welcome :)
Dominique needs to solve the system of equations by graphing. Which statement is true about the system?
2x-3y=12
y=-3/5x+2
A)Both equations are in slope-intercept form.
B)Neither equation is in slope-intercept form.
C)The first equation is y= 2/3x-4 when written in slope-intercept form.
D)The second equation is 3x+5y=10 when written in slope-intercept form.
Answers
Answer:
Option C) The first equation is y= 2/3x-4 when written in slope-intercept form
Step-by-step explanation:
we have
[tex]2x-3y=12[/tex] ----> First equation
The first equation is in standard form
Convert to slope intercept form
Isolate the variable y
[tex]3y=2x-12[/tex]
Divide by 3 both sides
[tex]y=\frac{2}{3}x-4[/tex]
[tex]y=-\frac{3}{5}x+2[/tex] ----> Second equation
The second equation is in slope intercept form
Convert to standard form
Multiply by 5 both sides to remove the fraction
[tex]5y=-3x+10[/tex]
[tex]3x+5y=10[/tex]
Verify each statement
case A) Both equations are in slope-intercept form.
False
Because only the second equation is in slope -intercept form
case B) Neither equation is in slope-intercept form
False
Because, the second equation is in slope -intercept form
case C) The first equation is y= 2/3x-4 when written in slope-intercept form
True (see the procedure)
case D)The second equation is 3x+5y=10 when written in slope-intercept form.
False
Because, the second equation is 3x+5y=10 when written in standard form
Which expression is equal to 7(1+3)
A.) 7+21
B.) 7+3
C.) 1+21
D.) 7+4
Answers
I would say D. 7+4 because it is the only one that makes since to me
A rectangular board is 1.8 meters long and 1.5 meters wide. What is the area of the board in square millimeters?
Answers
The area of the rectangular board is 2,700,000 mm².
Explanation:The subject of this question is Mathematics and it's likely suitable for a Middle School student. The question asks for the area of a rectangular board given specific length and width in meters, and requires the answer in square millimeters.
First, let's calculate the area of the rectangular board in square meters (m²) by multiplying the length and the width. So, 1.8 m * 1.5 m equals 2.7 m². Now, to convert the area from square meters to square millimeters (mm²), remember that 1 m² is equal to 1,000,000 mm². Therefore, the area of the board in square millimeters is 2.7 m² * 1,000,000 equals 2,700,000 mm².
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What is the sine value of pi over 6?
Answers
Answer:
The sine value of pi over 6 is:
[tex]\dfrac{1}{2}[/tex] or 0.5
Step-by-step explanation:
we have to find the value of [tex]sin\dfrac{\pi}{6}[/tex]
we know that value of [tex]sin\dfrac{\pi}{6}[/tex] is [tex]\dfrac{1}{2}[/tex]
Hence, the sine value of pi over 6 is:
[tex]\dfrac{1}{2}[/tex] or 0.5
Suppose you buy a movie for $12.98 the sales tax is 6% find the total cost
Answers
12.98*0.06=0.7788
Add the tax to 12.98, and you get 13.7588
The total cost is 13.76
hope this helps:)
Three-fourths of the yard is covered in grass but 4/5 of the grass has died. How much of the yard has dead grass?
Answers
multiply the 2 together:
3/4 x 4/5 = 12/20 = 3/5 has dead grass
To find the area of the yard that has dead grass, multiply three-fourths (the grassy area) by 4/5 (the dead portion). Simplifying (3/4) × (4/5) gives us 3/5, which means 3/5 of the yard has dead grass.
To determine how much of the yard has dead grass, we need to calculate the portion of the yard covered in grass and then the fraction of it that has died. Given that three-fourths of the yard is covered in grass, we represent this as 3/4. It is known that 4/5 of this grass has died, hence we multiply the two fractions: (3/4) × (4/5).
Performing the multiplication of these two fractions:
Multiply the numerators: 3 × 4 = 12.Multiply the denominators: 4 × 5 = 20.Combine the products to form the new fraction: 12/20.Now, simplify the fraction 12/20 by dividing both the numerator and denominator by their greatest common divisor, which is 4:
Divide the numerator: 12 ÷ 4 = 3.Divide the denominator: 20 ÷ 4 = 5.The simplified fraction is 3/5.Therefore, the correct answer is that 3/5 of the yard has dead grass.
In triangle ∆PQR, C is the centroid.
a. If CY = 10, find PC and PY
b. If QC = 10, find ZC and ZQ
c. If PX = 20, find PQ
Answers
Because C is the centroid,
PZ = ZR; QY = YR; and PX = XQ.
Part a.
If CY = 10, then PC = 2*CY = 20
PY = PC + CY = 20 + 10 = 30
Answer:
PC = 20
PY = 30
Part b.
If QC = 10, then ZC = QC/2 = 5
ZQ = ZC + QC = 5 + 10 = 15
Answer:
ZC = 5
ZQ = 15
Part c.
Given: PX = 20.
Because the median bisects side PQ, therefore PX = QX = 20
PQ = PX + QX = 40
Answer:
PQ = 40
There are two numbers, which sum up to 53. Three times the smaller number is equal to 19 more than the larger number. What are the numbers
Answers
the larger number = 3n-19
n + 3n - 19 = 53
4n = 53 + 19
4n = 72
n = 72/4
n = 18 ← smaller number
larger number = 53 - 18 = 35
Answer: 18 and 35.
Let x be the smaller number; then it'll be 3x = 19 + y
Set the first equation equal to x, then substitute the values and solve.
x + y = 53.
x = 53 - y.
3(53 - y) = 19 + y
x = 18; y = 35
140 - y = y
y = 70.
53 - 70 = -17
x = -17; y = 70.
-17 + 70 = 53;