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?
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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!
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!
