in the book that i'm using, i feel that they dont explain how to solve absolute value equations very well, so i'd like confirmation from an instructor to make sure i'm doing it correctly and that i have covered all scenarios
22 - |y+14| = 20
i first solve to get the inequality as the only thing on one side of the equation, then i know that there are 2 options
-|y+14| = 20 - 22
-|y+14| = -2
|y+14| = 2
1.
y+14 = 2
y = -12
2.a.
y + 14 = -2
y = -16
2.b. since the value inside the absolute value brackets can be negative, i mutliply everything within the absolute value brackets by negative 1
-y - 14 = 2
-y = 16
y = 16
i know that 2.a. and 2.b. actually are equal to each other, but is one way the correct way of doing absolute value equations?
_____________________________________________________________________________________________________
separately for inequalities, i did some googling and came across a site that said that if the greater than/less than sign is facing one way you have to do one thing and if it's the other way you can solve it by flipping the sign.
like i think for less than or equal to it said to solve it like this:
ax - b <= c, then -c <= ax - b <= c
and for greater than or equal to, it said for one solution you would solve it normally, then you find the 2nd solution by changing the direction of the inequality sign
rather than having to remember when to flip signs, i'm trying to think of a hard and fast rule i can apply to both absolute value equations and inequalities. again, could an instructor verify that what i'm doing is correct?
|2x - 5| <= 7
1.
2x - 5 <= 7
2x <= 12
x <= 6
2.a.
2x - 5 >= -7
2x >= -2
x >= -1
2.b.
-2x + 5 <=7
-2x <= 2
x >= -1
again, i know that 2.a. and 2.b. are equivalent, but is it correct that if i make the 7 negative, i then reverse the way the inequality is pointing? or is the correct way to do it the 2.b. way, or vice versa. i'm just very confused about inequalities
then lastly,
|-3x + 2 | > 7
1.
-3x + 2 >7
-3x > 5
3x < 5
x < (5/3)
2.a.
-3x + 2 < -7
-3x < -9
3x > 9
x > 3
2.b.
3x - 2 > 7
3x > 9
x > 3
i know that step 1 is always needed for the 3 examples i gave. i think i prefer to do step 2.b. for all equations because there is no flippng of signs, and it's consistent that you have to always make everything in the absolute value bracket negative. just need confirmation.
thank you very much!
how to solve absolute values and absolute value inequalities
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- Anurag@Gurome
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That's what you already did in 2.a.dhlee922 wrote:2.a.
y + 14 = -2
y = -16
2.b. since the value inside the absolute value brackets can be negative, i mutliply everything within the absolute value brackets by negative 1
-y - 14 = 2
...
i know that 2.a. and 2.b. actually are equal to each other, but is one way the correct way of doing absolute value equations?
2.a ---> (y + 14) = -2 <---- Multiply both sides by -1 --> -y - 14 = 2 <--- 2.b
2.b is completely redundant as it is a different version of 2.a
By definition of |x|,
- |x| = x if x ≥ 0 and |x| = -x if x < 0
- if (y + 14) ≥ 0, |y + 14| = (y + 14) = 2 ---> y = -12
if (y + 14) < 0, |y + 14| = -(y + 14) = 2 ---> y = -16
The same logic applies to all the other examples also.
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You're not doing anything wrong except that redundancy part.dhlee922 wrote:rather than having to remember when to flip signs, i'm trying to think of a hard and fast rule i can apply to both absolute value equations and inequalities. again, could an instructor verify that what i'm doing is correct?
But before remembering this hard and fast "rules" and "shortcuts", I'll suggest you to tackle these problems logically. Once you practice to do so there won't be any rules to remember.
The practical implication of absolute value is distance from zero on the number line. |x| means distance of x from zero on the number line.
Now, let us take your first example, |2x - 5| ≤ 7
What does this inequality mean?
It means that the absolute value of (2x - 5) is less than or equal to 7, i.e. maximum value of the distance of (2x - 5) from zero on the number line is 7.
Try to visualize the situation on the number line and you can quickly conclude that minimum value of (2x - 5) is -7 and maximum value of (2x - 5) is 7.
Hence, -7 ≤ (2x - 5) ≤ 7
Or we can write the inequality separately as...
- (2x - 5) ≤ 7 ------> Your equation number 1
and
(2x - 5) ≥ - 7 ------> Your equation number 2.a
From definition of absolute value we know that, if (2x - 5) ≥ 0, |2x - 5| = (2x - 5) and if (2x - 5) < 0, |2x - 5| = -(2x - 5)
Hence,
- If (2x - 5) ≥ 0 ---> (2x - 5) ≤ 7
If (2x - 5) < 0 ---> -(2x - 5) ≤ 7 ---> (2x - 5) ≤ -7
Follow the same logic with your last example, |-3x + 2| < 7 and let me know if you face any issue.
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Anurag@Gurome wrote:That's what you already did in 2.a.dhlee922 wrote:2.a.
y + 14 = -2
y = -16
2.b. since the value inside the absolute value brackets can be negative, i mutliply everything within the absolute value brackets by negative 1
-y - 14 = 2
...
i know that 2.a. and 2.b. actually are equal to each other, but is one way the correct way of doing absolute value equations?
2.a ---> (y + 14) = -2 <---- Multiply both sides by -1 --> -y - 14 = 2 <--- 2.b
2.b is completely redundant as it is a different version of 2.a
By definition of |x|,Hence, here
- |x| = x if x ≥ 0 and |x| = -x if x < 0
I hope you understand the possible two different scenarios by logic not by merely remembering the process.
- if (y + 14) ≥ 0, |y + 14| = (y + 14) = 2 ---> y = -12
if (y + 14) < 0, |y + 14| = -(y + 14) = 2 ---> y = -16
The same logic applies to all the other examples also.
thanks anurag for the replies. i'm obviously rusty in simple math skills, but i thought you always have to multiply both sides of the equation, but the 2.a. example only multiplies one side of the equation. is that what you always do for inequality questions?
i initially meant that when solving these i would not do both 2.a. and 2.b. I just found that 2a and 2b were 2 ways to think of it and i would choose one of them, and i believe your 2nd reply is similarly stating that 2b is the way to go based on your 2nd reply
i really appreciate you explaining the logic behind it as that has cleared things up.
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If |x| > k (where k is greater than 0) then x > k or x , -krkav wrote:|-3x + 2 | > 7
1.
-3x + 2 >7
-3x > 5
3x < 5
x < (5/3)
For this example why is x < (5/3) as opposed to x <(-5/3)
-3x+2 > 7
-3x> 5
(divide by negative 3)
x< -5/3
where did i go wrong? thanks!
So, if |-3x + 2 | > 7, then -3x + 2 > 7 or -3x + 2 < -7
Tackle each one separately:
-3x + 2 > 7
-3x > 5
x < -5/3
-3x + 2 < -7
-3x < -9
x > 3
So, x < -5/3 or x > 3
Cheers,
Brent
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The problem occurred when you went from -3x > 5 to 3x < 5rkav wrote:|-3x + 2 | > 7
-3x + 2 >7
-3x > 5
3x < 5
x < (5/3)
If we multiply both sides by -1, we reverse the inequality and get 3x < -5
Cheers,
Brent
Thank you Brent.Brent@GMATPrepNow wrote:If |x| > k (where k is greater than 0) then x > k or x , -krkav wrote:|-3x + 2 | > 7
1.
-3x + 2 >7
-3x > 5
3x < 5
x < (5/3)
For this example why is x < (5/3) as opposed to x <(-5/3)
-3x+2 > 7
-3x> 5
(divide by negative 3)
x< -5/3
where did i go wrong? thanks!
So, if |-3x + 2 | > 7, then -3x + 2 > 7 or -3x + 2 < -7
Tackle each one separately:
-3x + 2 > 7
-3x > 5
x < -5/3
-3x + 2 < -7
-3x < -9
x > 3
So, x < -5/3 or x > 3
Cheers,
Brent
So just to be clear one of the answers is x< negative 5/3?
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x is either less than -5/3 or greater than 3
Cheers,
Brent
Cheers,
Brent
hi, i have a follow up question regarding absolute values
when is |x-4| equal to 4-x?
so i start with 2 equations, first one is:
x-4 = 4-x
2x = 8
x = 4
then 2nd equation i have:
-(x-4) = 4-x
-x+4 = 4-x
these 2 equations are equal, so i thought you would have only one solution, but the book says the answer is x <= 4
how come i cant apply step 2.b. from the initial question at the top to this problem?
when is |x-4| equal to 4-x?
so i start with 2 equations, first one is:
x-4 = 4-x
2x = 8
x = 4
then 2nd equation i have:
-(x-4) = 4-x
-x+4 = 4-x
these 2 equations are equal, so i thought you would have only one solution, but the book says the answer is x <= 4
how come i cant apply step 2.b. from the initial question at the top to this problem?
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Your methods work well when you start with an equation or inequality. But here you are just asked a question about some expression. There is no equation given here, so they expect you to fall back on the definition of absolute value notation.
|(stuff)| means (stuff), when stuff is positive, and -(stuff) when stuff is negative, for example |4| is (4), but |-4| is -(-4).
In your example, |x-4| will equal 4-x whenever "stuff" is negative, ie in this case when (x-4) is negative. That happens when x<4.
The "zero" solution is interesting to think about too. I would keep my eye pealed for answers playing on the edge of any constraints I derived. You can use the answers to tip you off and then test the extreme values.
Cheers,
misterholmes
|(stuff)| means (stuff), when stuff is positive, and -(stuff) when stuff is negative, for example |4| is (4), but |-4| is -(-4).
In your example, |x-4| will equal 4-x whenever "stuff" is negative, ie in this case when (x-4) is negative. That happens when x<4.
The "zero" solution is interesting to think about too. I would keep my eye pealed for answers playing on the edge of any constraints I derived. You can use the answers to tip you off and then test the extreme values.
Cheers,
misterholmes
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