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How do you solve #abs(x-2)>x+4#?

Answer 1

Ethan Adkins

The solution is #x in (-oo,-1)#

This is solving an inequality with absolute values.

#x-2=0#, #=>#, #x=2#

There are #2# intervals to consider

#(-oo, 2)# and #(2,+oo)#

In the Interval #(-oo,2)#

#-x+2>x+4#

#=>#, #2x<-2#

#=>#, #x<-1#

As # x<-1# #in# the interval, the solution is accepted

In the Interval #(2,+oo)#

#x-2>x+4#

#=>#, #0>6#

There is no solution in the interval.

The solution is #x in (-oo,-1)#

graph{|x-2|-x-4 [-18.01, 18.02, -9, 9.01]}

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Answer 2

Jace Anderson

To solve the inequality (|x-2| > x+4), you need to consider two cases:

When (x-2) is non-negative ((x \geq 2)): In this case, the absolute value (|x-2|) is equal to (x-2). So the inequality becomes: (x - 2 > x + 4) Simplifying, we get: (-2 > 4)

This is a contradiction, which means there are no solutions in this case.
When (x-2) is negative ((x < 2)): In this case, the absolute value (|x-2|) is equal to (-(x-2)), which is (-x+2). So the inequality becomes: (-x + 2 > x + 4) Simplifying, we get: (2 > 2x + 4) (-2 > 2x) (-1 > x)

Therefore, the solution to the inequality is (x < -1).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

How do you solve #abs(x-2)&gt;x+4#?

How do you solve #abs(x-2)>x+4#?