Questions on Algebra: Absolute value answered by real tutors!

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: I have come to a point in solving : is this function one-to-one, where the absolute value of a = the absolute value of b. Can this be one to one?
For the record, the problem states: f(x)= |x| - 2
I assigned a and b, and equal them to each other.
|a| - 2 = |b| - 2 ; add two to both sides
|a| = |b|
therefore a = b? or not really?

No it does not! for instance we could take a=3 and b=-3, then

|3| = |-3|

Another way to say it is it's not one-to-one because,
for instance f(3) = f(-3) =  1 but 3 does not equal -3.   

The points (3,1) and (-3,1) are both on the graph and
a horizontal line goes through them both, so the graph does not pass the horizontal line test.

The graph looks like this:



And as you see, the horizontal lines cut it more than
once, so it cannot be one-to-one.



Edwin

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you are correct

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Hi, Hope I can help,
.
Solve
.
Absolute value will always make the numbers inside positive, absolute value is the distance away from "0" on the number line
.
|5| = 5, it is 5 units away from zero
.
|-5| = 5, it is also 5 units away from zero
.
, to get rid of the absolute value signs, we can say the left side equals 10, as well as (-10), since the absolute will make it positive, so we really have two equations
.
, and , now let us solve for "x"
.
First equation,
.
We need to move (-4) to the right side
.
= = , to find "x" we need to divide each side by "7"
.
= = , we can check by replacing "x" with "2" in the equation
.
= = = (True)
.
Now let us solve the second equation,
.
We need to move (-4) to the right side
.
= = , to find "x" we will divide each side by "7"
.
= = , we can check by replacing "x" with ()
.
= = = (True)
.
Your two answers are ( ), and ( )
.
If we replaced these two numbers into the original equation, they should work
.
replace with "2", = = = = ( absolute value means that it is "10" units from "0" )
.
replace with ( ), = = = = ( True ) ( Remember, the absolute signs will make any number positive, absolute value is just how far the number is from "0", (-10) is 10 units from "0"
.
Your two answers are
.

.

.
Hope I helped, Levi

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hm i think its 100

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Remember, the absolute value of ANY number is ALWAYS positive. So the absolute value of is ALWAYS positive. Algebraically, this means that


Since the right side is negative (and the "less than" sign wants more negative numbers), this means that there are NO solutions.

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Perimeter of a Triangle -----> , EQN 1
where ---->
Subst. in EQN 1:

, ANSWER
Thank you,
Jojo

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consecutive integers are n,n+1 and n+2
so n+n+1+n+2=99 then 3n+3=99 then 3n=96 then n=32
Solution: 32,33,34

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by definition:
that's what absolute value means.
if the number is negative, then the absolute value of the number is minus the number. this makes it a plus.
if it's a plus, then the absolute value of the number is the number without any modifications because it's plus already.

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3n-2

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The number of U.S. teenagers in the age category "15-19-years olds" during any year between 1980 and 1998 can be modeled approximately by
T=332*|x-11.5| + 17000
where T is the number of "15-19-years-olds" in thousands and x is the number of years since January 1, 1980 (U.S. Census Bureau, Current Poplulatioin Reports.) Use the model to determine the year in which the number of 15-19-year olds in the U.S. was 18,700,000 (18,700 thousands).
The book comes up with the answer of 1986 and 1996,
------------------------------------------------------
18,700 = 332*|x-11.5| + 17000
-----
332*|x-11.5| = 1700
|x-11.5| = 5.12..
x-11.5 = 5.12 or x-11.5 = -5.12...
x = 16.62 or x = 6.38
--------------------------
1980 + 6.38 = 1986
1980 + 16.62 = 1996
=======================
Cheers,
Stan H.

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Start with the given equation


Subtract 17,000 from both sides.


Divide both sides by 332 (this is an approximate value)



Break up the absolute value (remember, if you have , then or )

or Set the expression equal to the original value 5.12048 and it's opposite -5.12048




Now lets focus on the first equation


Add 11 to both sides


Combine like terms on the right side


So rounding to the nearest whole number, we get


This means that 6 years from the beginning (which is 1980), the population of 15-19 yr olds was 18,700,000.

So in 1986 (which is 6 years away from 1980), the population of 15-19 yr olds was 18,700,000.

---

Now lets focus on the second equation



Add 11 to both sides


Combine like terms on the right side


So rounding to the nearest whole number, we get



This means that 16 years from the beginning (which is 1980), the population of 15-19 yr olds was 18,700,000.

So in 1996 (which is 16 years away from 1980), the population of 15-19 yr olds was 18,700,000.



=========================================


Answer:

So we got the solutions (after rounding to the nearest whole number) and . These solutions represented the number of years after 1980.

So in 1986 and in 1996 the population of 15-19 year olds was 18,700,000

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Start with the given equation


Subtract 3 from both sides.


Break up the absolute value (remember, if you have , then or )

or Set the expression equal to the original value 5 and it's opposite -5




Now lets focus on the first equation


Add 1 to both sides


Combine like terms on the right side


Divide both sides by 4 to isolate t



Divide

---

Now lets focus on the second equation



Add 1 to both sides


Combine like terms on the right side


Divide both sides by 4 to isolate t



Reduce




=============================================

Answer:


So the solutions are:

and

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-[3+4]+[-5]+[6-10]=
-|7| + 5 + |-4| =
-7 + 5 + 4 =
-7 + 9 = 2
The absolute value is distance and so it is always positive.

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5|r+3|> 5
Solving an absolute value inequality problem is similar to solving an absolute value equation.
Start by isolating the absolute value on one side of the inequality symbol, then follow the rules below:
Because the symbol is greater than (>), we use the word OR.
In this sample, the number on the right side of the symbol > is bigger than 0.
First thing to do is ISOLATE the absolute value symbol.
We do this by dividing both sides by 5.
So, 5/5 = 1
We now have this question:
|r + 3| = 1
We now set up two cases:
r > 1 or r < -1
Remove the value from inside the absolute value solving each case individually.
CASE ONE:
r + 3 > 1
r > -3 + 1
r > -2
=========
CASE TWO:
r + 3 < -1
r < -3 - 1
r < -4
Final answer:
r > -2 or r < - 4

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abs l 1/2x +2l = l 3/4x -2 l Find the solution set for the equation
-------------------
(1/2)x +2 = (3/4)x-2 or (1/2)x + 2 = -(3/4)x+2
(1/4)x = 4 or (5/4)x = 0
x = 16 or x = 0
----------------------
Cheers,
Stan H.

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This is a problem we had on an algebra test 2y-3=absolute value of -15 pretty sure the answer is nine, but my math teacher says 9,-6
-------------------
The absolute value of -15 is +15, so the answer would be 9.
-----------------------
If the problem states the AV of (2y-3) = -15, then both 9 and -6 would fit. Check the statement of the problem again.

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Find the possible value or values of r in the quadratic equation r2-7r-8=0
---
Factor to get:
(r-8)(r+1) = 0
Since the product is zero, one of these factors must be zero:
So, r=8 or r=-1
===================
Cheers,
Stan H.

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The denominator is always positive.
The numerator is always positive or zero.
Therefore, there is no value of x such that the left hand side would be less than zero.
No solution.


By definition,
for all Y.
.
.
.
Graphical verification.

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Assume y=c is a solution. There are 2 cases:
(1) c is positive, in this case |c|=c and we get,
|c| - 4 = 3c
c-4=3c
2c=-4
c=-2
we assumed c is positive so this solution is not true,
(2) c is negative, in this case |c|=-c and we get,
|c| - 4 = 3c
-c-4=3c
4c=-4
c=-1
we assumed c is negative so this answer is true.

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To solve an absolute value equation, isolate the absolute value on one side of the equal sign, and establish two cases:
Case One:
Set the expression inside the
absolute value symbol equal to
the other given expression.
Case Two:
Set the expression inside the
absolute value symbol equal to
the negation of the other given
expression
Your question:
2|d + 3| = 8
We want to isolate the absolute value first.
We divide both sides by 2 to get this done.
|d + 3| = 8/2
|d + 3| = 4
We now establish two cases as written above.
Case One:
d + 3 = 4
d = 4 - 3
d = 1
Case Two:
d + 3 = -4
d = -4 - 3
d = -7
Now, you must check. The two cases create "derived" equations. These derived equations may not always be true equivalents to the original equation. Consequently, the roots of the derived equations MUST BE CHECKED in the original equation so that you do not list extraneous roots as answers.
We plug our answers of d = 1 and d = -7 in the original absolute value equation given to see which will create the same answer on both sides of the equation.
Let d = 1
2|1 + 3| = 8
2|4| = 8
2(4) = 8
8 = 8...It checks!!
We now let d = -7
2|-7 + 3| = 8
2|-4| = 8
2(4) = 8
8 = 8...It also checks!!
So, our final answer is: d = 1 and d = -7
Did you follow?

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Start with the given inequality


Break up the absolute value (remember, if you have , then or )

or Break up the absolute value inequality using the given rule




Now lets focus on the first inequality


Start with the given inequality


Add 5 to both sides


Combine like terms on the right side


Divide both sides by -3 to isolate x (note: Remember, dividing both sides by a negative number flips the inequality sign)



Reduce


Now lets focus on the second inequality


Start with the given inequality


Add 5 to both sides


Combine like terms on the right side


Divide both sides by -3 to isolate x (note: Remember, dividing both sides by a negative number flips the inequality sign)



Divide



----------------------------------------------------

Answer:

So our answer is

or


So the solution in interval notation is: (] [)


So the solution in set builder notation is:




Here's the graph of the solution set




Note:
There is a closed circle at which means that we're including that value from the solution set.


Also, there is a closed circle at which means that we're including that value from the solution set.

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just to the right of where you typed ] there should be another key that has a vertical line over a backslash.
the vertical line is the absolute value symbol |
the backslash looks like this: \
the | is upper case.
the \ is lower case.
it may not look like one single line. it might look like two smaller vertical lines separated by a space in the middle.
hit the key and it should be a solid line on your screen.
on my keyboard it's the last key on the right below the backspace key and above the return key.
-----
your problem states that -2 * |7d| = -14
divide both sides of the equation by -2.
equation becomes:
|7d| = 14
-----
since 7d is within the absolute value sign, then d can be either positive or negative. the absolute value of d will always be positive.
-----
id d is positive, then |7d| = 7d
if d is negative, then |7d| = -7d
-----
solve for when d is positive.
then solve for when d is negative.
-----
when d is positive,
7d = 7
d = 1
when d is negative,
-7d = 7
-d = 1
d = -1
-----
your answer is:
d +/-1
-----
to prove, substitute in original equation
first for when d is positive
d = 1
-2*|7*1| = -2*|7| = -2*(+7) = -14
next for when d is negative
d = -1
-2*|7*(-1)| = -2*|-7| = -2*(-(-7)) = -2*(+7) = -14
-----
whether or not d was +1, or -1, the answer was the same because the absolute value of a positive number is a positive number, and the absolute value of a negative number is a positive number.
-----
good luck with finding your | key.

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Solve the following absolute value inequality: I am not quit sure how to do this problem with a fraction. Thank you

Multiply both sides by 







RULES for removing absolute value bars in inequalities.
(Assume A is a non-negative number)

1.  becomes 
2.  becomes 
3.  becomes 
4.  becomes 

Your problem is case 1

 becomes 

Solve for x in the middle



Add 2 to all three sides:





In interval notation:

(-6, 10)

Edwin

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once you get x=-8 you plug it into the equation.
|x-7|=|2x+1|
|(-8)-7|=|2(-8)+1|
|-15|=|-16+1|
|-15|=|-15|
15=15

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x is less than -3 or greater than 3: y is less that -1 or greater than 1.

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Like this you mean...

if so, then absolute value problems are really two problems in one: a positive and a negative solution.
.
.
.
Positive solution:


Negative solution:



d=16 and d=-16 are the two solutions.

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-(8 +x)=7
-8-x =7
x=-8 -7 =-15
x = -15

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-(8)+X=7
-8+X=7
X=7+8
X=15 ANSWER.
OR:
-(8+X)=7
-8-X=7
-X=7+8
-X=15
X=-15 ANSWER.

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Please help me solve the following question:
Solve and write in interval notation for the solution set:
The absolute value of x+2 greater than or equal to 4

The rules for removing the absolute value bars in inequalities
are

1.  can be rewritten as 

2.  can be rewritten as 

3.  can be rewritten as 

4.  can be rewritten as 

Your problem:


is case 4.

 

can be rewritten as



Add  to both sides in both parts:



      

<==========@-----------------------@=========>
 -9 -8 -7 -6 -5 -4 -3 -2 -1  0  1  2  3  4  5

   

Edwin

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When dealing with absolute values, you need to know that what comes out of the absval sign is always positive. Hence, you need to consider that. This is what I mean:
The right hand side is = 12, so the left hand side must = 12 OR -12
So, set the left hand side = to both 12 and -12.
2x-4 = 12, and solve for x
Add 4 to both sides to get 2x = 16, and divide both sides by 2 to get x by itself:
x=16/2 = 8. So, 8 is one of your answers.
Now, set the left hand side = -12:
2x-4 = -12
Add 4 to both sides to get 2x = -8, and divide both sides by 2 to get x by itself:
x=-8/2 = -4. So -4 is the other answer.
So, this problem has two solutions, 8 and -4.
Good luck,
JoeC

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Start with the given equation


Break up the absolute value (remember, if you have , then or )

or Set the expression equal to the original value 22 and it's opposite -22




Now lets focus on the first equation


Add 5 to both sides


Combine like terms on the right side


Divide both sides by 3 to isolate x



Reduce

---

Now lets focus on the second equation



Add 5 to both sides


Combine like terms on the right side


Divide both sides by 3 to isolate x



Divide





So the solutions to are:

and

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Start with the given equation


Break up the absolute value (remember, if you have , then or )

or Set the expression equal to the original value 12 and it's opposite -12




Now lets focus on the first equation


Add 4 to both sides


Combine like terms on the right side


Divide both sides by 2 to isolate x



Divide

---

Now lets focus on the second equation



Add 4 to both sides


Combine like terms on the right side


Divide both sides by 2 to isolate x



Divide





So the solutions to are:

and



Notice if we graph and (just set each side equal to y and graph), we get


Graph of (red) and (green)

and we can see the two graphs intersect at and . So this verifies our answer.

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if i understand you correctly, the problem is:
-----
y = |x+2| - 1
-----
the equation presented was already solved.
there was nothing else that could be done to simplify it further.
not that i could see.
all that was needed to do was plot some points and graph it.
i graphed it as is.
the answer will go negative because the -1 is outside the absolute value symbols. i believe that's what you meant when you showed it.
plot some points yourself from x = -5 to x = 5 to see what's happening.
graph of this equation looks like this:

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if you want to know what number 20 is 40% of, then you need to make an equation that says 40% of x = 20
.4*x = 20
x = 20/.4
x = 50
-----
to test, take 40% of 50 = 20
----
you needed to do 20 / 40%

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Well, you're not totally off base, perhaps only half off!
Solve for n:
Remove the the absolute-value bars, but note you will get two equations as a result!
or Now you solve these two equations, the first of which you have done correctly.
First one:
Add 6 to both sides.
Divide both sides by 5.

or
Second one:
Add 6 to both sides.
Divide both sides by 5.

or

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Graph of

To find the coordinates, simply plug in values of "x" to find values of "y" (which will give you ordered pairs)

Let's find "y" when x=-1


Start with the given equation


Plug in


Evaluate the absolute value of -1 to get 1


Multiply


So one point is (-1,3)


------------------------------------------------------


Let's find "y" when x=0


Start with the given equation


Plug in


Evaluate the absolute value of 0 to get 0


Multiply


So another point is (0,0)


------------------------------------------------------


Let's find "y" when x=1


Start with the given equation


Plug in


Evaluate the absolute value of 1 to get 1


Multiply


So another point is (1,3)


------------------------------------------------------

So we have the points (-1,3), (0,0) and (1,3)


So here's the graph along with the points

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if f(x) = x, then
f(3) = 3
f(5) = 5
you substitute the x with a value of x and solve.
in your problem,
f(x) = 3*|x|
set y = f(x) so the equation becomes y = 3*|x|.
now pick some x values and you'll get some y values.
example:
if x = 3, f(x) = y = 3 * |3| = 3 * (3) = 9
if x = 1, f(x) = y = 3 * |1| = 3 * (1) = 1
if x = 0, f(x) = y = 3 * |0| = 3 * (0) = 0
if x = -1, f(x) = y = 3 * |-1| = 3 * (1) = 1
if x = -3, f(x) = y = 3 * |-3| = 3 * (3) = 9
-----
whether x is negative or positive, the y value which is the same as f(x) will always be positive.
-----
plot points from x = -5 to x = + 5 at least and you'll see the shape of the curve.
you may already see it from the points already plotted here.

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x-7=x+7x
-------------
Collect terms
x-7 = 8x
Subtract x from each side
-7 = 7x
x = -1

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5x–4y=1 multiply this equation by 2 & add them.
-10x+8y=-3
10x-8y=2
-----------------------
0x+0y=-1 indicates there is no solution.

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Hi, Hope I can help
.
WHICH EQUATIONS OF LINES PASS THROUGH 0,6
Y=6X+5
Y=5X+6
Y=5X+3
Y=4X+6
.
Points are given as (x,y)
.
Point (0,6)(x,y), it means "x" = "0", "y" = "6"
.
To find if point (0,6) is a point on a line, all you do is replace "x" and "y" in the equations with the point, in this case (0,6)
.
Replace "x" with "0", and "y" with "6", in all of the equations
.
(0,6), = = = (False)
.
(0,6), = = = ( True) ( Point (0,6) is a point on the line )
.
(0,6), = = = ( False )
.
(0,6), = = = ( True ) ( Point (0,6) is a point on the line )
.
Here are the lines on a graph
.
Blue Line =
Redish Brown Line =
Green Line =
Purple Line =
.

.
The line equations, and , have (0,6) as a point
.
Hope I helped, Levi

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"The answers to an absolute value problem are graphed below. What was the equation?"
The graph had -12 and 120 graphed. I know that I find the medain number between the two, but then what should I do?
---------------------------
The median number is (-12+120)/2 = 108/2 = 54
----------
The distance from the median to an end point is 120-54 = 66
------------
Absolute value is an expression of distance:
|x-54| = 66
This means "The distance each of your numbers is from 54 is 66.
One of the numbers is 66 above 54.
The other number is 66 below 54.
================
Cheers,
Stan H.

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Start the saem way you do when there is an equal sign. When evaluating absolute values, you need to break the equation into 2 parts.

breaks into
and
Now solve each part for s

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Find all the real values for x so that (5-x) < 2. Write your answer in a sentence.
**** ( ) signifies absolute value
-------------------------------------
|5-x| < 2
Rewrite as follows:
-2 < 5-x < 2
Subtract 5 along the line to get:
-7 < -x < -3
Multiply thru by -1 to get:
3 < x < 7
--------------
Cheers,
Stan H.