9514 1404 393
Explanation:
In general, a system of equations has a solution that satisfies all of the equations. That is, the solution set is the intersection (and) of the solution sets of the individual equations in the system. Equations involving the absolute value function are no different.
That being said, you need to consider the meaning of an equation involving the absolute value function.
The function itself is piecewise defined:
[tex]\displaystyle |x|=\left\{ {x, x\ge0} \atop {-x, x<0}} \right.[/tex]
So, any equation involving the absolute value function automatically resolves to two equations, each with its own condition on the function value.
__
Example 1:
|x+1| > 3
is equivalent to the two disjoint conditional equations ...
(x +1) > 3 and (x +1) ≥ 0 . . . . OR
-(x +1) > 3 and (x +1) < 0
the first of these has the solution x > 2; the second of these has the solution x < -4. The solution set of this equation is the OR of the two solution sets:
x < -4 or x > 2
__
Example 2:
|x +1| < 3
is equivalent to the two disjoint conditional equations ...
(x +1) < 3 and (x +1) ≥ 0 . . . . OR
-(x +1) < 3 and (x +1) < 0
The first of these has the solution -1 ≤ x < 2, and the second of these has the solution -4 < x < -1. Again, the solution set of this equation is the OR of the two solution sets. However, we find we can write that union as a single compound inequality:
-4 < x < 2
__
When an absolute value equation involves more than one absolute value function, then the equation will probably resolve into multiple equations, each defined on its own domain. Sometimes keeping those domains straight can be tedious.
Above, we have considered inequalities. If you have an equation, you need to consider that it will likely resolve to two equations. The solution set will be the OR of the solutions to those two equations.
Example 3:
|x +1| = 3
is equivalent to the two conditional equations ...
(x +1) = 3 and (x +1) ≥ 0 . . . . OR
-(x +1) = 3 and (x +1) < 0
The first of these has the solution x = 2; the second has the solution x = -4. The solution set is the OR of these two solutions:
x = -4 or x = 2
__
The "and" condition restricts the domain of each of the individual pieces of the equation. The "or" condition applies to the collection of solutions that may exist in each of those restricted domains.
help i will mark brainlist if i cannn
Answer:
B
Step-by-step explanation:
Answer:
a. 1 and 1/4.
Step-by-step explanation:
I simplified it.
Please mark me brainliest!
what is next in the sequence ? 17, 13, 9, 5
Answer:
1
Step-by-step explanation:
It is being subtracted by four everytime
15 time the difference of 14 and 6
Answer:
120
Step-by-step explanation:
14-6=8x15=120
3x²+2x²+x-2 divided x+2
Answer:
3x^2-4x+9 remainder -20/x+2
Step-by-step explanation:
someone help me find x please!
total cost spent at a play avries directly with how many tickets are purchased. if any buys 5 tickets and spends 30 dollars. how much would 25 tickets cost
Answer:
$150
Step-by-step explanation:
5 x 6 = 30
25 x 6 = 150
Answer:
If it is a solution of equations, then plug your solution into the equations you are working on to make sure that all equations are satisfied by your solution.
Step-by-step explanation:
I hope that this was helpful.
How would you simplify 3/16 times 6/11 before finding the product
answere:
You can't simplify
explanation:
there is no need to simplify because 3/16 and 6/11 is the simplest form.
[tex]the \: angle \: of \: measure \: \frac{31\pi}{6} \: lies \: in \: which \: quadrant[/tex]
Answer:
3rd quadrant
Step-by-step explanation:
First we just convert the angle from radians to degrees[tex]\frac{31\pi radians }{6}*\frac{180degrees}{\pi radians} =\frac{31*180}{6} degrees=930degrees[/tex] Now that's too big, all this means is if we start rotating from the positive y-axis in a circle we will cross the starting point 2 times, 2 full circles;[tex]930degrees-360degrees=570degrees\\570degrees-360degrees=210degrees[/tex] Now in which quadrant it 210 degrees?0 degrees to 90 degrees is 1st quadrant90 degrees to 180 degrees is 2nd quadrant180 degrees to 270 degrees is 3rd quadrant270 degrees to 360 degrees is 4th quadrantSo our answer is the 3rd quadrant.What is the equation of a line that is parallel to x−2y=−4 and passes through the point (0, 0)?
Enter your answer in the box.
Answer:
The equation of a line that is parallel to x−2y=−4 and passes through the point (0, 0) will be: [tex]x-2y=0[/tex]
Step-by-step explanation:
Given the line
[tex]x-2y=-4[/tex]
[tex]x+4 = 2y[/tex]
[tex]y=\left(\frac{1}{2}\right)x+2[/tex]
[tex]m_1=\frac{1}{2}[/tex]
so, the slope of the line parallel to the given line would have
the same slope as [tex]\frac{1}{2}=m[/tex], and passed through the point (0, 0).
Therefore, the equation of the required line is given by
[tex]y = mx+b[/tex]
[tex]y=\left(\frac{1}{2}\right)x+0[/tex]
[tex]y=\left(\frac{1}{2}\right)x[/tex]
[tex]2y=x[/tex]
[tex]x-2y=0[/tex]
Therefore, the equation of a line that is parallel to x−2y=−4 and passes through the point (0, 0) will be: [tex]x-2y=0[/tex]
Write an equation in slope intercept form using the information below.
We want the equation in the form y = mx + b.
We are given that m = -7.
The y-intercept (our b) is 8.
Here it is:
y = -7x + 8
Use substitution to find an exponential function, y= a - b" that passes through
(2, 16) and (5, 1.024).
Answer:
[tex]y = 4^x[/tex]
Step-by-step explanation:
Given
[tex](2,16)[/tex]
[tex](5,1,024)[/tex]
Required
Determine the exponential function [tex]y = a.b^x[/tex]
In [tex](2,16)[/tex]
[tex]x=2[/tex] and [tex]y = 16[/tex]
Substitute these values in [tex]y = a.b^x[/tex]
[tex]16 = a.b^2[/tex] --- (1)
In [tex](5,1,024)[/tex]
[tex]x = 5[/tex] and [tex]y = 1024[/tex]
Substitute these values in [tex]y = a.b^x[/tex]
[tex]1024 = a.b^5[/tex] --- (2)
Divide (2) by (1)
[tex]\frac{1024}{16} = \frac{a.b^5}{a.b^2}[/tex]
[tex]\frac{1024}{16} = \frac{b^5}{b^2}[/tex]
[tex]64 = \frac{b^5}{b^2}[/tex]
Apply second law of indices
[tex]64 = b^{5-2[/tex]
[tex]64 = b^3[/tex]
Express 64 as an exponent of 3
[tex]4^3 = b^3[/tex]
[tex]4 = b[/tex]
[tex]b = 4[/tex]
Substitute 4 for b in (1)
[tex]16 = a.b^2[/tex]
[tex]16 = a * 4^2[/tex]
[tex]16 = a * 16[/tex]
[tex]16 = 16a[/tex]
Solve for a
[tex]a=\frac{16}{16}[/tex]
[tex]a = 1[/tex]
Substitute 1 for a and 4 for b in [tex]y = a.b^x[/tex]
[tex]y = 1 * 4^x[/tex]
[tex]y = 4^x[/tex]
The exponential function is given by y = 100(0.4)ˣ
An exponential function is in the form:
y = abˣ
where y, x are variables, a is the initial value of y and b is the factor.
At point (2, 16):
16 = ab² (1)
At point (5, 1.024):
1.024 = ab⁵ (2)
Dividing equation 2 by equation 1:
b³ = 0.064
b = 0.4
16 = a(0.4)²
a = 100
The exponential function is given by: y = 100(0.4)ˣ
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PLS Help .Kennedy Middle School is having a Halloween celebration. Students visit each classroom and solve math problems to get candy. For each problem they solve correctly, they earn 8 pieces of candy. Jack's teacher gave him 4 pieces of candy to start with. Jack's goal is to get at least 60 pieces of candy. Graph on the number line the number of problems Jack will need to solve correctly to meet his goal.
Answer:
Ok im too too sure but I think that it is the one with the filled in dot -->
Step-by-step explanation:
Jose has $4. His mom gives him $2 each time he does his homework. He is saving up to
buy a game that costs $68. How many times does he have to complete his homework in
order to reach this goal?
Help please if you can
Answer: mutiply x by itself to get y
Step-by-step explanation:
a company’s stock was selling at 35$ a share. a month later it was selling at 14$ a share. what is the percent loss?
Answer:
60%
Step-by-step explanation:
its right
If a company’s stock was selling at 35$ a share. a month later it was selling at 14$ a share then the percent of loss is 60%.
What is Percentage?percentage, a relative value indicating hundredth parts of any quantity.
Given that a company’s stock was selling at 35$ a share.
After a month later it was selling at 14$ a share.
We need to find the percent of loss.
Original share price=35
new share price=14
We calculate the percentage by finding the difference and dividing the difference with the original share price.
35-14/35
0.6
Now multiply this 0.6 with 100 as it is a hundredth part.
0.6×100
60%
Hence, if a company’s stock was selling at 35$ a share. a month later it was selling at 14$ a share then the percent of loss is 60%.
To learn more on Percentage click:
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HELP PLease! 40 points!!!!
Answer:
D. 480 ftStep-by-step explanation:
Let the sides of triangle be a, b and c
Perimeter
P = a + b+ c = 40New sides
12a, 12b and 12cNew perimeter
Pn = 12a + 12b + 12c = 12(a+b+c) = 12*40 = 480 ftCorrect choice is D
Determine the degree of each monomial.
Answer:
Step-by-step explanation:
4= 0 degree
2z= 1 degree
4r^2st^3= 6 degree
3xyz^2= 4 degree
What is 83 1/3% of 30
I need help ♀️What acronym help you decide order to multiply Binomials ?
Answer:
FOIL
Step-by-step explanation:
" is just a mnemonic. It helps some people to remember all the steps needed to multiply two binomials. It "makes sense" because it's methodical and gives the right answer.
A 50-foot flagpole near an office building casts a shadow
that is 12.5 feet long. At the same time, the office building
casts a shadow that is 50 feet long. How tall is the office
building?
Step-by-step explanation:
Here are some hints: "At the same time,"means you need to set up a proportion (two ratios equal each other).
The shadow is always on the ground.
A person and a flagpole always stands up straight, like a wall (vertical height)
Underline key words and numbers. Draw a picture for sure.
|
|X 5.5 ft. |
|____________ |____
flagpole=50 feet person's shadow=4 feet
shadow
Since it is easier to work with one type of measurement, change the person's height to feet only. The person is 5feet, 6inches Note: 12 inches= 1 foot, so 6 inches= 1/2 a feet and 6 inches= .5 feet
URGENT WILL GIVE BRAINLIST, please answer these 3 questions
Answer:
Q.8:use the equation [tex]y-y_1=m(x-x_1)[/tex] [tex]y-(-4)=-6(x-(7))\\y+4=-6x+42\\y=-6x+38[/tex]Q.9:Find slope with [tex]slope=\frac{y_2-y_1}{x_2-x_1}[/tex] substitute and solve[tex]slope=\frac{(1)-(-3)}{(3)-(1)} =2[/tex] then as follows[tex]y-y_1=m(x-x_1)\\y-(1)=2(x-(3))\\y-1=2x-6\\y=2x-5[/tex] Q.10:Find slope [tex]slope=\frac{(12)-(2)}{(2)-(7)}=-2[/tex] then[tex]y-y_1=m(x-x_1)\\y-(2)=-2(x-(7))\\y-2=-2x+14\\y=-2x+16[/tex]A pool is in the shape of a rectangular prism. On Monday, water was pumped out of the pool at a constant rate, starting at 12:00 p.m. At 12:15 pm, the water in the pool was 45 inches deep. At 12:25 p.m., the water in the pool was 41 inches deep. Write an equation that represents y, the depth of the water (in inches), after x minutes.
Answer:
y = 51-0.4x
Step-by-step explanation:
From the question, we can see that the amount of water left in the pool was 45 inches deep within 12:15-12:00 = 15 minutes
However, within 10 minutes there was a loss of 45-41 = 4 inches
So what this means is that per minute, the amount of loss expected is 4 inches/10 minutes = 0.4 inches per minute
what we are saying is that a loss of 4 inches depth of water is needed per minute
So we want to write an equation, representing the depth after x minutes
So what we have here is that the total inches at present will be;
45 + 0.4(15) = 45 + 6 = 51 inches
The pool was 51 inches deep initially before the pumping out began
That will be;
y = 51-0.4x
where is the depth and x is the number of minutes
Answer:
24 inches
Step-by-step explanation:
Find the slope of the question y=5
Answer:
0,5
Step-by-step explanation:
slope: 0
y-intercept:(0,5)
PLEASE HELP MEEEEEEE
Answer:
triangle BFD
75 degrees
Using the slope formula, find the slope of the line through the points (0,0) and (4,12).
Answer:
3
Step-by-step explanation:
slop of given point is 3
What is the inverse of f(x) = sqft x+ 3
Answer:
[tex]f^{-1} (x)=x^{2} -3[/tex]
Step-by-step explanation:
Given the function
[tex]f\left(x\right)=\sqrt{x+3}[/tex]
We can find the inverse function by exchanging x and y, and then solve for y.
[tex]y=\sqrt{x+3}[/tex]
Replace x with y.
[tex]x=\sqrt{y+3}[/tex]
square both sides
[tex]x^2=y+3[/tex]
[tex]y=x^2-3[/tex]
Hence,
[tex]f^{-1} (x)=x^{2} -3[/tex]
I need help very important
Answer:
∠ EFG = 83°, ∠ GCE = 97°
Step-by-step explanation:
Since FE and FG are tangents to the circle then
∠ FGC and ∠ FEC are right angles
The sum of the angles in quadrilateral CEFG = 360°
Sum the 4 angles and equate to 360
3x + 11 + 90 + 5x - 23 + 90 = 360, that is
8x + 168 = 360 ( subtract 168 from both sides )
8x = 192 ( divide both sides by 8 )
x = 24
Thus
∠ EFG = 3x + 11 = 3(24) + 11 = 72 + 11 = 83°
∠ GCE = 5x - 23 = 5(24) - 23 = 120 - 23 = 97°
Shaq invested $900 into a savings account that pays annual simple interest. At the end of 9 years, he earned $461.70 in interest. What is the interest rate on the savings account? Round to the nearest tenth of a percent.
Answer:
Interest = $900 * x * 9 years with "x" = interest rate
If interest = 461.70 then
461.70 = 900 * 9 * x
x = 461.70 / 8,100
x = interest rate = .057 = 5.7%
Double Check
Interest rate of 5.7 per cent equals
.057 * 900 = $51.30 per year
Nine Years = 9 * 51.30 equals 461.70
So, it's correct!!!
Step-by-step explanation:
Which statement correctly compares the function shown on this graph with the function y =5x+5.
A. The function shown on the graph has a greater rate of change, but a lower starting point.
B. The function shown on the graph has a greater rate of change and a higher starting point.
C. The function shown on the graph has a smaller rate of change, but a higher starting point
D. The function shown on the graph has a smaller rate of change and a lower starting point.
D. The function shown on the graph has a smaller rate of change and a lower starting point. Sorry if incorrect i hope i helped.
omar buys a pack of 8 towels for 24.00 find the unit price in dollar per towel
Answer:
um 3
Step-by-step explanation:
whats 24 divided by 3 mate? 3
i mean all u need is a calculator im dumb as hell and i know that...
hope...this helps you :) have a great day