# Solve for w |6w+2|-6>8

|6w+2|-6>8
Write |6w+2|-6>8 as a piecewise.
To find the interval for the first piece, find where the inside of the absolute value is non-negative.
6w+2≥0
Solve the inequality.
Subtract 2 from both sides of the inequality.
6w≥-2
Divide each term by 6 and simplify.
Divide each term in 6w≥-2 by 6.
6w6≥-26
Cancel the common factor of 6.
Cancel the common factor.
6w6≥-26
Divide w by 1.
w≥-26
w≥-26
Simplify -26.
Cancel the common factor of -2 and 6.
Factor 2 out of -2.
w≥2(-1)6
Cancel the common factors.
Factor 2 out of 6.
w≥2⋅-12⋅3
Cancel the common factor.
w≥2⋅-12⋅3
Rewrite the expression.
w≥-13
w≥-13
w≥-13
Move the negative in front of the fraction.
w≥-13
w≥-13
w≥-13
w≥-13
In the piece where 6w+2 is non-negative, remove the absolute value.
6w+2-6>8
To find the interval for the second piece, find where the inside of the absolute value is negative.
6w+2<0
Solve the inequality.
Subtract 2 from both sides of the inequality.
6w<-2
Divide each term by 6 and simplify.
Divide each term in 6w<-2 by 6.
6w6<-26
Cancel the common factor of 6.
Cancel the common factor.
6w6<-26
Divide w by 1.
w<-26
w<-26
Simplify -26.
Cancel the common factor of -2 and 6.
Factor 2 out of -2.
w<2(-1)6
Cancel the common factors.
Factor 2 out of 6.
w<2⋅-12⋅3
Cancel the common factor.
w<2⋅-12⋅3
Rewrite the expression.
w<-13
w<-13
w<-13
Move the negative in front of the fraction.
w<-13
w<-13
w<-13
w<-13
In the piece where 6w+2 is negative, remove the absolute value and multiply by -1.
-(6w+2)-6>8
Write as a piecewise.
{6w+2-6>8w≥-13-(6w+2)-6>8w<-13
Subtract 6 from 2.
{6w-4>8w≥-13-(6w+2)-6>8w<-13
Simplify -(6w+2)-6>8.
Simplify each term.
Apply the distributive property.
{6w-4>8w≥-13-(6w)-1⋅2-6>8w<-13
Multiply 6 by -1.
{6w-4>8w≥-13-6w-1⋅2-6>8w<-13
Multiply -1 by 2.
{6w-4>8w≥-13-6w-2-6>8w<-13
{6w-4>8w≥-13-6w-2-6>8w<-13
Subtract 6 from -2.
{6w-4>8w≥-13-6w-8>8w<-13
{6w-4>8w≥-13-6w-8>8w<-13
{6w-4>8w≥-13-6w-8>8w<-13
Solve 6w-4>8 when w≥-13.
Solve 6w-4>8 for w.
Move all terms not containing w to the right side of the inequality.
Add 4 to both sides of the inequality.
6w>8+4
6w>12
6w>12
Divide each term by 6 and simplify.
Divide each term in 6w>12 by 6.
6w6>126
Cancel the common factor of 6.
Cancel the common factor.
6w6>126
Divide w by 1.
w>126
w>126
Divide 12 by 6.
w>2
w>2
w>2
Find the intersection of w>2 and w≥-13.
w>2
w>2
Solve -6w-8>8 when w<-13.
Solve -6w-8>8 for w.
Move all terms not containing w to the right side of the inequality.
Add 8 to both sides of the inequality.
-6w>8+8
-6w>16
-6w>16
Divide each term by -6 and simplify.
Divide each term in -6w>16 by -6. When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.
-6w-6<16-6
Cancel the common factor of -6.
Cancel the common factor.
-6w-6<16-6
Divide w by 1.
w<16-6
w<16-6
Simplify 16-6.
Cancel the common factor of 16 and -6.
Factor 2 out of 16.
w<2(8)-6
Cancel the common factors.
Factor 2 out of -6.
w<2⋅82⋅-3
Cancel the common factor.
w<2⋅82⋅-3
Rewrite the expression.
w<8-3
w<8-3
w<8-3
Move the negative in front of the fraction.
w<-83
w<-83
w<-83
w<-83
Find the intersection of w<-83 and w<-13.
w<-83
w<-83
Find the union of the solutions.
w<-83 or w>2
The result can be shown in multiple forms.
Inequality Form:
w<-83 or w>2
Interval Notation:
(-∞,-83)∪(2,∞)
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Solve for w |6w+2|-6>8