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How do you solve #abs(4x+2)<6#?

Answer 1

Hazel Anglin

See below

Use definition of absolute value

#absx=x# if #x>=0# and #absx=-x# if x<0

In our case

first case: #abs(4x+2)=4x+2# if #4x+2>=0# but this last it's the same that: #4x>=-2# or #x>=-1/2#

If #x>=-1/2# then #abs(4x+2)=4x+2<6# or #4x<4# or #x<1#

Then #abs(4x+2)<6# if #x in [-1/2,1]#

Second case: #abs(4x+2)=-(4x+2)# if #4x+2<0#. But this it's the same that #x<-1/2#

If #x<-1/2# then #abs(4x+2)=-(4x+2)=-4x-2<6# or #-8<4x# or #-2< x#

then #abs(4x+2)<6# if #x in (-2, -1/2)#

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Answer 2

Abigail Allen

To solve the inequality ( |4x + 2| < 6 ), you can follow these steps:

Set up two inequalities: [ \begin{align*} &4x + 2 < 6 \ &4x + 2 > -6 \end{align*} ]
Solve each inequality separately: [ \begin{align*} &4x + 2 < 6 \ &\Rightarrow 4x < 4 \ &\Rightarrow x < 1 \ \ &4x + 2 > -6 \ &\Rightarrow 4x > -8 \ &\Rightarrow x > -2 \end{align*} ]
Combine the solutions: The solution is the intersection of the two sets: [ -2 < x < 1 ]

So, the solution to ( |4x + 2| < 6 ) is ( -2 < 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(4x+2)&lt;6#?

How do you solve #abs(4x+2)<6#?