|x+16|<8

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

x+16≥0

Subtract 16 from both sides of the inequality.

x≥-16

In the piece where x+16 is non-negative, remove the absolute value.

x+16<8

To find the interval for the second piece, find where the inside of the absolute value is negative.

x+16<0

Subtract 16 from both sides of the inequality.

x<-16

In the piece where x+16 is negative, remove the absolute value and multiply by -1.

-(x+16)<8

Write as a piecewise.

{x+16<8x≥-16-(x+16)<8x<-16

Simplify -(x+16)<8.

Apply the distributive property.

{x+16<8x≥-16-x-1⋅16<8x<-16

Multiply -1 by 16.

{x+16<8x≥-16-x-16<8x<-16

{x+16<8x≥-16-x-16<8x<-16

{x+16<8x≥-16-x-16<8x<-16

Move all terms not containing x to the right side of the inequality.

Subtract 16 from both sides of the inequality.

x<8-16

Subtract 16 from 8.

x<-8

x<-8

Find the intersection of x<-8 and x≥-16.

-16≤x<-8

-16≤x<-8

Solve -x-16<8 for x.

Move all terms not containing x to the right side of the inequality.

Add 16 to both sides of the inequality.

-x<8+16

Add 8 and 16.

-x<24

-x<24

Multiply each term in -x<24 by -1

Multiply each term in -x<24 by -1. When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

(-x)⋅-1>24⋅-1

Multiply (-x)⋅-1.

Multiply -1 by -1.

1x>24⋅-1

Multiply x by 1.

x>24⋅-1

x>24⋅-1

Multiply 24 by -1.

x>-24

x>-24

x>-24

Find the intersection of x>-24 and x<-16.

-24<x<-16

-24<x<-16

Find the union of the solutions.

-24<x<-8

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