Math Problem?

Question

How do you solve absolute value inequalities? Please give an example.

Answer 1

That's a complicated question, luv. Absolute value inequalities are not that hard to solve, but you have to remember a couple of things.

1) Any time you're dealing with absolute value, you're really dealing with two different equations. You get one of them when the quantity in the absolute value signs is positive, and the other when it is negative.

2) The solution to an absolute value inequality - when there is one - will most commonly take the form or either a conjunction or a disjunction. In other words, it'll either be something like

-1 < x < 6 (conjunction)

or it'll be like

x < -1 or x > 6 (disjunction).

Easy example:

|x| < 5

Part 1: Assume x is positive. Then the absolute value sign does nothing, meaning x >= 0 (x is positive) AND x < 5. The conjunction of these two will be

0 <= x < 5

Part 2: Assume x is negative. Then the absolute value sign reverses the sign of x, meaning x <= 0 (x is negative) AND -x < 5

-x < 5

x > -5 (dividing by -1 reverses the inequality).

The conjunction of these is -5 < x <= 0

The disjunction of the two solutions is

-5 < x < 5

Which can easily be checked.

If the expression in the absolute value sign is more complex, it doesn't really matter. The solution proceeds the same way. Take the first cut with what's in there being positive, drop the absolute value signs, and solve, then find the conjunction of the results. Then take the second cut with what's in there being negative, reverse the sign, and solve. Then take the conjunction of the last two. The final solution will be the disjunction of parts 1 and 2.

Hope this helps.

Answer 2

when solving abs value inequalities I came up with a little saying for my students
Al Gore was vice president of our land...shortened to Gore Land
if the symbol is > or >= or = use "or" when solving the iequality
if the symbol is < or <= use "and"

how does gore land help? G for > and e for = with the word or (makes GORE)
and L for < with the word and (makes Land)

so if you have abs value (x+3) > 7
set up two inequalities x+3 > 7 OR x+3<-7
and solve so x>4 OR x< -10

also if you have abs value (x+3) = 7
set up two inequalities x+3 = 7 OR x+3=-7
and solve so x=4 OR x= -10


if you have abs value (x+3)<7
set up two inequalities x+3 <7 AND x+3 >-7
and solve so x<4 AND x>-10 which can also be written as
-10 < x <4

hope that helps

Answer 3

Similar to how you solve regular abosulte value equations:

2 > |x|

2 > x, 2 > -x

Simplfy the second equation by dividing by "-1"

2 > x, -2 < x

Therefore:

-2 < x < 2

Also:

|x + 3| > 1

x + 3 > 1, -x - 3 > 1

Simplifys:

x > -2, -x >4 ---> x > -2, x < -4

Therefore:
-4 > x > -2

Hope this helps!

Answer 4

You have two answers, most of the time.

So let's say you have a very simple equasion:

|x| > 5

There are two possibilities, either X can be greater than 5
(a number that would fit this problem would include 6, 8, 100,000)
OR, because negative numbers count backwards, X would be LESS THAN -5 (-6, -8, -100,000).

So you have to remember to do two things,
(a) do two equasions, one with X and one with -x
(b) SWITCH the inequality sign when you do the "negative x" equasion.

Answer 5

Ex: I -6 I = 6
Ex: I 6 I = 6
No matter what the number inside is it will always be positive.

Answer 6

y = |x|, x = +y, x = -y

example: y = |x| + 3

|x| = y - 3
x = y - 3, x = -(y - 3) = 3 - y