# Questions on Algebra: Rational Functions, analyzing and graphing answered by real tutors!

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You can put this solution on YOUR website!
you have to solve for y in order to place the equation into y = mx + b form.
3x = 9 + 3y
subtract 9 from both sides of the equation to get:
3x - 9 = 3y
divide both sides of the equation by 3 to get:
3x/3 - 9/3 = y
simplify to get:
x - 3 = y
if x - 3 = y, then y = x - 3
y = x - 3 is the y = mx + b form.
m is equal to 1 (coefficient of x)
b is equal to -3 (constant term)
m is the slope of the equation.
b is the y-intercept of the equation.
this means the slope of your equation is equal to 1 and the y-intercept of your equation is equal to -3.
graph of your equation looks like this:

Question 817024: how do I help solve for x and y. I only get 0=7??
4x-5y=7
-4x+5y=7

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4x-5y=7
-4x+5y=7 times -1 --> 4x-5y = -7
-------------
The equations are inconsistent.
4x-5y can't be both 7 & -7.
---
If you graph them, you get parallel lines.
No intersection, no solution.

Find all asymptotes, intercepts and graph. you must also find all the critical points and inflections points (if any)
f(x) = x^3-3x^2-10x/x^2+5x+6

Answer by Edwin McCravy(9716)   (Show Source):
You can put this solution on YOUR website!

f(x) =

Factor the numerator:     Factor the denominator

x³-3x²-10x                x²+5x+6
x(x²-3x-10)              (x+3)(x+2)
x(x+3)(x+2)

f(x) =

Since (x+3) is a factor of the denominator but not
the numerator, there is an asymptote where x+3=0,
or at x=-3, which is the equation of the vertical
asymptote, where there is a non-removable discontinuite.
Since (x+2) is a factor of both denominator and numerator,
there is a removable discontinuity where x+2=0, at x=-2.

We may cancel the (x+2)'s as long as we also state that
x≠2

f(x) = , where x≠2

So we graph

y = , leaving a hole at x=2

There is a vertical asymptote at x=-3
Since the degree of the numberator is 1 more than the degree
of the denominator, there is no horizontal asymptote, but there
is an oblique (or slant) asymptote, which we find by long
division:

We have to multiply the numerator out and add +0 to divide:

y = ,

x- 8+
x+3)x²-5x+ 0
x²+3x
-8x+ 0
-8x-24
34

Since the fraction  approaches 0 as x gets large,
the graph of f(x) must approach the line y=x-8, which is the
equation of the oblique (slant) asymptote.

We get the y-intercept by setting x = 0

y =  = 0

So the y-intercept is (0,0)

We get the x-intercepts by setting y = 0

0 =

0 = x(x-5)
x=0;  x-5=0
x=5

So the x-intercepts are (0,0) and (5,0)

We plot the asymptotes and the intercepts:

Now we find any relative extrema points by
finding the derivative and setting it = 0

y =
Multiply the top out:
y =
Use the quoptient formula for the derivative:
y' =
y' =
y' =
Setting that = 0 to find relative extrema:
= 0
x²+6x-15 = 0
Unfortunately that doesn't factor, so we must

x = -3 ± 2V6
Approximating:  x=-7.90 and x=1.90
Substuting those in y, we get approximately
y=-20.8 and y=-1.20

Relative extrema candidates are approximately (-7.90,-20.8)
and (1.90,-1.20)

To find out whether they are relative maximums or minimums,
or any inflection points, we must find the second derivative:

y' =
Use the quotient formula:
y" =
y" =
y" =
y" =
y" =
y" =
y" =

Substituting x=-7.90, y" comes out negative,
therefore the point (-7.90,-20.8) is a relative
maximum, since the curvature is downward

Substituting x=1.90, y" comes out positive,
therefore the point (1.90,-1.20) is a relative
minimum, since the curvature is upward

To find any inflection points we set y"=0

= 0
48 = 0
A contradiction so there are no inflection points.

So we draw the graph:

What a terribly long and messy problem!

Edwin

Question 816648: One and one-half ounces of a medication are required for a 120-pound adult. At the same rate, how many additional ounces of medication are required for a 180-pound adult?

You can put this solution on YOUR website!
1.5/120 = m/180
m = 180(1.5/120)
m = 2.25
---
dm = 2.25 - 1.5
dm = 0.75
---
a 180 lb adult requires 3/4 of an ounce more medication than a 120 lb adult
---
Solve and graph linear equations:
www.sooeet.com/math/linear-equation-solver.php
---
---
Convert fractions, decimals, and percents:
http://www.sooeet.com/math/fraction-decimal-percent.php

Question 816114: What are the vertex, axis of symmetry, maximum or minimum value, domain and range for
Y=-x^2-4x+3

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-x^2 - 4x + 3
---
the above quadratic equation is in standard form, with a=-1, b=-4, and c=3
---
to solve the quadratic equation, plug this:
-1 -4 3
---
the quadratic vertex is a maximum at ( -2, 7 )
---
the axis of symmetry is: x = -2
---
the domain is the real numbers
---
the range is y <= 7
---
Solve and graph linear equations:
www.sooeet.com/math/linear-equation-solver.php
---
---
Convert fractions, decimals, and percents:
http://www.sooeet.com/math/fraction-decimal-percent.php

Question 816121: List the transformations for
y=-(x-2)^2+4
This has to do with quadratic functions and transformations

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The - sign in front of the parentheses reflects the graph of the parent function over the x-axis.

The -2 inside the parentheses shifts the graph of the parent function to the right two units.

The +4 outside the parentheses translates the graph of the parent function up 4 units.

For more help from me, visit: www.algebrahouse.com

Question 815936: how to solve this problem and can you show the work please 1/x + 1/x+5 = x+6/x+5

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+ =
multiply each term by x(x+5), this gets rid of the denominators and we have
(x+5) + x = x(x+6)
x + 5 + x = x^2 + 6x
2x + 5 = x^2 + 6x
Arrange as a quadratic equation on the right
0 = x^2 + 6x - 2x - 5
x^2 + 4x - 5 = 0
Factors to
(x+5)(x-1) = 0
Two solutions
x = -5
x = 1
When you try these solutions in the original problem, you can see that only
x = 1 is a valid solution, x=-5 gives us denominators of 0

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is an expression. Expressions are not solved. Please re-post your question and include the actual instructions for the problem.

Question 815775: 7. A rancher wants to build a rectangular enclosure for his new heard of thestrals. He wants to maximize the total area for his new heard, and has 5280ft of fencing in which to build the enclosure.What is the maximum area the rancher could enclose with his fencing? Round to the nearest foot.
Bonus: How many acre's is that? (round to the nearest acre)

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Hi,
Re reply, Maximum Area is 1,742,400 ft^2 0r 40 acres
5280ft of fencing
A Square yields maximum Area. = 1320ft a side
A = s^2 = = 1,742,400 ft^2
maximum Area = 1,742,400 ft^2
Note: 43,560 ft^2 = 1 Acre
maximum Area(in Acres) = = 40 Acre

Question 815407: Identify the holes and sketch the graph : (x^2+5x+4)/(-4x-8)
* None of the factored terms could be canceled, so I couldn't identify any holes. But are there any? Also, I searched for the graph of this function and used my graphing calculator, but they looked different.

Found 2 solutions by Edwin McCravy, richwmiller:
Answer by Edwin McCravy(9716)   (Show Source):
You can put this solution on YOUR website!
f(x) =

f(x) =

There are no holes (i.e., removable discontinuities).
This happens in rational functions only when the numerator
and denominator have a common factor which has a zero.

This has a vertical asymptote whose equation is -4x-8=0
-4x=8
x=-2

It has no horizontal asymptote since the degree of the numerator
is larger than the degree of the denominator, but since the
numerator's degree exceeds the denominator's degree by 1, there
is an oblique (slant) asymptote:

To find the oblique asymptote we divide the denominator into the
numerator.  This will be difficult and will involve fractions unless
we factor out the common factor out -4 in the denominator

f(x) =

f(x) =

x+3-
x+2)x²+5x+4
x²+2x
3x+4
3x+6
-2

f(x) = [x+3-]

Since the fraction  approaches zero as x gets
very large, the curve must approach the line y = (x+3), thus
that is the equation of the oblique (slant) asymptote.

We draw the asymptotes:

We get the intercepts:

= 0

x²+5x+4 = 0
(x+4)(x+1) = 0
x+4-0; x+1=0
x=-4;   x=-1

So the x intercepts are (-4,0) and (-1,0)

The y intercept is

f(0) =  =  =

The y intercept is (0,)

We plot the intercepts

Then we can sketch the curve passing through those intercepts
and approaching the asymptotes:

Edwin

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To find the asymptote not a hole as Edwin reminded me.
set the denominator to zero
-4x-8=0
-4x=8
x=8/-4
x=-2
There is an asymptote at x=-2

Question 814463: The Wireless cafe charges \$5.40 for 3/4 of an hour of internet access. How much money does the Wireless Cafe per hour?
You can put this solution on YOUR website!
3/4 x= \$5.40
x=\$5.4 *4/3
x=\$7.20

Question 814456: Use the Rational Zero Test to list all possible rational zeros of f.
f(x)=x^3-4x^2-4x+16

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since the leading coefficient is 1, we only look at the constant terms factors
16:
1, -1, 2, -2, 4, -4, 8, -8, 16, -16
in fact toe actual zeros are -2, 2, and 4

Question 815092: Hi I've been trying to find out what to do and how to do and I'm stuck so I'm hoping you can help me
Simplify the right-hand side of the equation
What is f of x equal x square plus seven x plus ten over x square subtract seven x subtract eighteen
It looks like F(x) = x^2+7x+10
---------------------- x^2-7x-18
And please show every step that will help so much thank you

Found 2 solutions by Edwin McCravy, josgarithmetic:
Answer by Edwin McCravy(9716)   (Show Source):
You can put this solution on YOUR website!
f(x) =

Since the degree of the numerator and denominator are both 2,
the horizontal asymptote has the equation

y =

y =

y = 1

Let's draw the horizontal asymptote y = 1 (in green)

This function f(x) is not defined when the denominator = 0.
It either has a vertical asymptote there or else it
has a hole in the curve.

We set the denominator = 0.

x²-7x-18 = 0
(x-9)(x+2) = 0
x-9=0;  x+2=0
x=9;    x=-2

So the function is not defined at either x=9 or x=-2.

The domain of f(x) is (-oo,-2)U(-2,9)U(9,oo)

Next we find out if there is a vertical asymptote or
a "hole in the curve" (removable discontinuity)
at x=9 and x=-2

Factor the numerator x²+7x+10 as (x+5)(x+2)
We have already factored the denominator x²-7x-18 as (x-9)(x+2).

f(x) =

Now we may ONLY cancel the (x+2)'s if we specify that x is not
equal to -2, for f(x) is not defined at x=-2 or at x=9.

But the fact that we have a factor (x-2) in the numerator and
also an (x-2) factor in the denominator tells us there is a
removable discontinuity at x=-2.  And since there is no (x-9)
factor in the numerator tells us that there is a vertical
asymptote at x=9.

So we draw the vertical asymptote (also in green) which has
the equation x=9

Now if we cancel the (x+2)'s we will get a function that we will
call g(x).  g(x) is exactly like f(x) except it will have a value at
x=2 whereas f(x) does not have a value there.

So the graph that removes the dicontinuity (plugs up the hole) is

g(x) =  has an asymptote at x=9

Let's first draw g(x), which does not have a hole:

In fact when x=-2 we have g(-2)===

But we don't want g(x), we want f(x), so we must put a hole in
the curve at the point (-2,), for that is a removable
discontinuity.

Edwin

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Are you asking for a simplification of ?

Factorize the numerator and denominator. Look for common factors which are equivalent to a factor of 1. Just be aware than if you cancel these, the meaning of the simplified form will be changed.

Numerator: (x+2)(x+5)
Denominator: (x+2)(x-9)

The function can be rewritten, .
BEFORE simplification, you must understand that the function has a skipped point at and an asymptote at .

The simplification is to cancel the apparant as a factor of 1:
.
You will note that I changed the name of this function. The discontinuity at is now gone, so the h(x) IS continuous there, but the asymptote at remains for h(x) as it does for F(x).

Question 813455: x^3-2x^2+5x-10
You can put this solution on YOUR website!
First, please provide the instructions for a problem. What are you/we supposed to do with this expression?

Second, please post your problem in an appropriate category. What you posted is a polynomial, not a rational function.

Problems clearly posted in an appropriate category will get the quickest responses.

Question 813636: State the intercepts and asymptotes for f and f-1
f(x)= (4^x-2)-3

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First, please put parentheses around exponents, especially if they are more than just a positive integer or variable. What you posted meant:

but I think you meant:

Problems that are clearly posted are more likely to get a quick response.

To find the y-intercept of f(x), make the x equal to zero:

Y-intercept: (0, -47/16)

To find the x-intercept(s), make the y be zero:

X-intercept: (, 0)

Since f(x) is an exponential function and since exponents can be any number, there are no vertical asymptotes. There is a horizontal asymptote. As x approaches negative infinity the exponent of approaches negative infinity, too. And as the exponent approaches negative infinity value of approaches zero. And since approaches zero, f(x) approaches -3. So:
Horizontal asymptote: y = -3

Finding these for the inverse is easy since for the inverse, the x's and y's have traded places. So the inverse's intercepts will be the function's intercepts with the x's and y's trading places and the inverse's asymptote will be the function's asymptote with the x's and y's trading places:

function inverse
Y-intercept: (0, -47/16) X-intercept: (-47/16, 0)
X-intercept: (, 0) Y-intercept: (0, )
Horizontal asymptote: y = -3 Vertical asymptote: x = -3

Question 814140: f(x)=x-7/x^2-4x-12
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First, please include the instructions for the problem. What are you/we supposed to do with this function?

Second, please use parentheses around numerators and denominators, especially if they are more than just a positive integer or variable. What you posted means:

while I suspect that you meant:
f(x)=(x-7)/(x^2-4x-12) or

Question 814175: Hi, I'm just confused with how to draw this rational function.
y= (x^2)/(x^2+2x-8)
I found the vertical asymptotes which are x= -4 and x= 2. I don't know how to sketch the graph from here. Thanks for the help in advance.
Lauren

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Here's the plot:

I guess you would just pick x's and then find the y's

Question 814061: Simplify x^2-25/x^2-3x-10
You can put this solution on YOUR website!
firstly (x^2-25) is known as the difference of perfect squares and can be written as (x+5)(x-5) or (x-5)(x+5) doesnt really matter..
with questions like this you need to factorise both numerator and denominator normally and cancel out a common factor

COMMON FACTOR OF (x-5)
which can then be broken down to because x/x has the same base and by indice laws anything with the same base in division you take the powers away from each other

Question 812958: Find the reciprocal of the rational expression.
(y^2 - 3y + 2)/ (y^2 - 4)

Question 813604: Hello, thank you for taking the time to answer my question! It is:
List the values of x (aka theta) in which y=secx I need the answer in radians.

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Using the half angle formula, find the exact value of
---------
x can be any value, but the secant is undefined for
x = pi/2 + n*pi, n = 0,1,2,3... radians.

Question 813624: state any restrictions on the variable. if no restrictions, say no restrictions.
(x-4)/(x+6)=(-9/6)

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Hi,
| Cannot divide by Zero ...so x definitely cannot be -6
Cross Multiplying to solve

6x-24 = -9x - 54
15x = -30
x = -2

Question 813609: I have a pretty interesting problem today:
I am told to use a graphing calculator and enter this function: y = tan x cot x
I know that tan and cot individually are the opposites of each other. When I put the above equation into a graphing calculator, I just get a straight horizontal line at the y = 1 However, I am told this graph is overlooking something, but am not sure what. I thought it might be the asymptotes, but I don't know. So, what is wrong/being overlooked by the calc in the creation of the graph?

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Hi,
y = tan x cot x Or y = tans x/tan x
Yes y =1, However, Note: at x = 0 , tan x = 0
θ radians sin θ cos θ tan θ
0° 0 0 1 0
30° π/6 1/2 √3/2 √3/3
45° π/4 √2/2 √2/2 1
60° π/3 √3/2 1/2 √3
90° π/2 1 0 ─

Question 813580: Find the area enclosed by the curve, y=25- x^2 and the straight line, y=x+13

Found 2 solutions by Alan3354, richwmiller:
You can put this solution on YOUR website!
Find the area enclosed by the curve, y=25- x^2 and the straight line, y=x+13
-------------
The 2 points of intersection are at x = -4 and x = 3.
y=25- x^2
INT(x) = 25x - x^3/3
INT(3) = 75 - 9 = 66
INT(-4) = -100 + 64/3 = -236/3
Area under parabola to x-axis = 66 + 236/3 = 434/3
-------------
Area under line = 87.5
--> 343/6 sq units

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integral_(-4)^3 (12-x-x^2) dx = 343/6=57.1667

Question 812955: Find the reciprocal of the rational expression
y^2 - 3y + 2/y^2-4

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The reciprocal of is . So you just flip the fraction to get this final answer

8 - 2z/9 x 18/3z - 12

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Hi,
If I am understanding Your Question correctly

= -5 1/3

Question 812905: Not sure about how to do this 1.
Find the value of the following expression
-2s - t - 6/-s + 4 + t
s = - 1 t = -6

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Hi,
s = - 1
t = -6
Find the value of the following expression
-2s - t - 6/-s + 4 + t
If as submitted there are no parenthesis:
-2(-1) - (-6) - 6/-(-1) + 4 + (-6)
2 + 6 -6 + 4 - 6 =

Question 812761: what is the y and x intercept of the graph function f(x)=-.20(x-7)^2+15
You can put this solution on YOUR website!
The y-intercept is the point where the graph crosses the y-axis,
which is the line ,
or the y-coordinate of that point, since we know that the point has .
We know it is only one point, because a function has at most one value of for each value of ,
so for , there is just one value of .
In other words, the graph of a function cannot cross a vertical line (like ) more than once.
--> --> --> --> -->
You would know if the answer is expected to be given as
or as .

The x-intercept is the point (or points) where the graph crosses the x-axis,
which is the line ,
or the x-coordinate of that point, since we know that the point has .
--> -->-->-->--> --> or .
The approximate values, are and
Again, you would know how you are expected to format the answer.

Question 812746: http://i.imgur.com/xx42CKl.png
Hey everyone!
Not sure how to do this, can I do it without a calculator or without graphing?
Thanks!

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Those are the simultaneous equations, x-3y=-1 and 2x-6y=2.

You should be able to judge something about the slopes of those lines just by looking/inspecting them. That should tell you something about a solution.

Question 812096: Please help me solve this word problem: a rectangular space of 252 square feet is allocated for living and dining areas in a house. Find the width of the square living area given that the width of the dining area is 9 feet.
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It would not make sense for the total rectangular area to be 9 feet wide, wide a very small 9 foot by 9 foot living room on one end, and a very long 9-foot wide dining area attached, like this:
That would be an easy answer, but a ridiculous floor plan.

What they have in mind looks like this:
The living area is a square with sides measuring . The dining room measures by 9 feet.
The total combined living plus dining area is a rectangle measuring by .
You want to find .

From there on, I would suggest factoring. It gets you from here to the answer in a few short steps.

If you cannot or will not do factoring, you can transform that equation into a more familiar form, and solve it without factoring.
-->-->
Then you could solve by completing the square, or apply the quadratic formula.
I'll show you both ways further down.
In the meantime, I will try to convince you that factoring often makes math easier.

The factoring required for this problem is easy, but it is good practice because, in math, factoring never completely goes away. It keeps coming back. When you are done with the math chapter on polynomials and factoring, you may think that you can forget it. (I did). Forget that idea. The need for factoring will keep coming up, over, and over, for as long as you have a math class.

Back to the problem.
The starting equation, says that there are two numbers, and , that area units apart, and multiply to give .
What are the factors of 252? You can tell that it is even, so is a factor. You may even realize that it is a multiple of because is a multiple of .
You realize that it is a multiple of and because the digits add up to .
If you divide by , or divide by twice , you get , so that tells you that is also a factor, and that
.
However, you do not need to figure out all that to solve the problem. You just need two numbers, differing by 9, whose product is 252.
You can start trying numbers in order.

5 is not a factor but 252 divided by 6 is 42, so

So far the pairs of factors differ in a lot more than 9,
but the differences are getting smaller.
7 and 8 do not evenly divide 252, but 9 does, and

10 and 11 do not work, but 252 divides by 12 to give 21, so
and so we have found the two factors
and are the width and length of the whole rectangle, and the living area is a square with sides measuring .

Solving the quadratic equation by completing the square:
If a quadratic equation has solutions that are rational numbers, it is easy enough to solve it by factoring (although not always as easy as done above).
Otherwise you can do it without formulas by "completing the square."
Starting from , you could realize that the left side of that equation is part of
,

So either -->-->--> ,
or -->-->--> , which does not make sense as a room length because it is a negative number.

A quadratic equation of the form ,
if it has any real solutions, the solutions are given by

That applies to , where
, , and .
So

-->
Since cannot be the measurement of the side of the living area in feet, the only solution that makes sense, so and the living area is a square with sides measuring .

Question 812132:
What is the expansion of (2x + 3) ^4 ?

You can put this solution on YOUR website!

What is the expansion of (2x + 3) ^4 ?
-------------
Multiply it and see.
=
---------
Or use Pascal's triangle.

Question 811860: 5n^2 - 20 m^2+4p
--------- x --------- = ?
m+2p 2m

Answer by Edwin McCravy(9716)   (Show Source):
You can put this solution on YOUR website!
=

Nothing cancels, so all you can do is
FOIL out the top and distribute the bottom

Edwin

Question 811675: If f(x)= 5/2x^2-4 and g(x)=3x+1 find f of g(x)
You can put this solution on YOUR website!
If f(x)= (5/2)x^2-4 and g(x)=3x+1 find f of g(x)
-----------------
f[g(x)] = f[3x+1] = (5/2)(3x+1)^2-4 = (5/2)(9x^2+6x+1)-4
----
= (45/2)x^2 + 15x - (3/2)
===========================
Cheers,
Stan H.

Question 811632: Graph inequality.

1/4y≤|x-1|

Answer by Edwin McCravy(9716)   (Show Source):
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y ≤ |x-1|

Multiply both sides by 4

y ≤ 4|x-1|

Draw the boundary graph of y = 4|x-1|

It's vertex is when what's between the | |'s equals 0

x-1 = 0
x = 1

Substitute in y = 4|x-1|
y = 4|1-1|
y = 4|0|
y = 4(0)
y = 0

So the vertex is (1,0)

We get a point on each side, let x=0, y = 4|0-1| = 4|-1| = 4(1) = 4
let x=2, y = 4|2-1| = 4|1| = 4(1) = 4

Plot points vertex (1,0) and points on each side (0,4) and (2,4)

Draw the graph of the boundary  y = 4|x-1|.  We draw it solid. not
dotted, because the original inequality was ≤, not < , so the
points on the boundary are solutions.

Test a point, say the origin (0,0) in the original inequality,
to see if it's a solution.

y ≤ |x-1|
(0) ≤ |0-1|
0 ≤ |-1|
0 ≤ 1

That's true.  The origin is a solution and therefore all points
on the same side of the graph that the origin is on are also
solutions. So we shade below the graph.

Edwin

Question 811469: Write a rational function with a vertical asymptote x=5 and horizontal asymptote y=1/2. I have no clue where to begin.
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Imagine the degree of numerator is equal to that of the denominator, and that one of the expression or factors in the denominator is (x-5).

Try something simple, . Seeing the way the denominator is chosen, x=5 will be a vertical asymptote. This will NOT give the horizontal asymptote wanted; so try a change.

Still the same x=5 vertical asymptote. Also there is now x=0 vertical asymptote. Note what would happen as x is increased without bound... The x in numerator and denominator would cancel as 1, and the x-5 in numerator and denominator would cancel as 1, and the expression would approach (1/2), being as horizontal asymptote.

Question 811452: Solve the equation for X
-4/x-1=7/2-x+3/x+1

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.
Is that what you have? If not, do you have, ?
Guessing that you have the second equation, multiply left and right sides of the equation by (x-1)(2-x)(x+1), which is the simplest common denominator, and simplify; and continue solving according to what you already have learned.

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The second STEP will result in
and third step may be

Question 811214: -5 + (4 - 2x) -8x + 5
(3x + 1) - (-3x - 7) - 9x
THANK YOU!~

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combine like terms

combine like terms

Question 810983: Good day, I would like help with finding the intervals of increase and decrease of this function and its local minimum and maximum if it has them. Since it is a rational function I used the quotient rule to derive it but I don't think I'm getting it right. It is y = x/(x-1)^2.
Thank you.

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I see two ways to calculate the derivative, and I like that,
because I know I make mistakes, so if both answers agree I'm reassured,
and if they don not agree, I try to find and fix errors.
(I will not say how it went this time).
This is what I ended up with:

Using the quotient rule

The other way:
.
So is another expression for the function,
and it could be useful for derivative calculation and more.
Now I will calculate the derivative of

With both answers in agreement, now I am pretty sure that
is the derivative.
changes sign at and at .
For , where ,
so and the function is increasing at that point,
and at all points with .
The denominator of changes sign at ,
which we know is a vertical asymptote,
so changes sign at .
For and the function starts decreasing after , (but the function does not change sign there).
The numerator of changes sign at ,
so for and the function is decreasing.
Since the function decreases for ,
increases for ,
and for the function exists, with ,
the function has a local minimum at .
, and zooming in

Question 810844: What value of x will make the rational function -x/x+1 undefined?

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What value of x will make the rational function -x/x+1 undefined?
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It's the value that makes the denominator zero.

Question 810847: How many horizontal asymptotes does the rational function f(x)=-2x^2/x^2+1 have?

Found 2 solutions by richwmiller, erica65404:
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the asymptote is when x^2+1=0
what will make x^2+1=0
If x^2=-1
if x=i

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-2 is the horizontal asymptote but it appears to be asking the amount of horizontal asymptotes so the answer would be 1.

Question 810826: 4-(-6)/-5