|x+23|<1

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

x+23≥0

Solve the inequality.

Multiply both sides of the equation by 3.

x+2≥0⋅(3)

Remove parentheses.

x+2≥0⋅(3)

Multiply 0 by 3.

x+2≥0

Subtract 2 from both sides of the inequality.

x≥-2

x≥-2

In the piece where x+23 is non-negative, remove the absolute value.

x+23<1

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

x+23<0

Solve the inequality.

Multiply both sides of the equation by 3.

x+2<0⋅(3)

Remove parentheses.

x+2<0⋅(3)

Multiply 0 by 3.

x+2<0

Subtract 2 from both sides of the inequality.

x<-2

x<-2

In the piece where x+23 is negative, remove the absolute value and multiply by -1.

-x+23<1

Write as a piecewise.

{x+23<1x≥-2-x+23<1x<-2

{x+23<1x≥-2-x+23<1x<-2

Solve x+23<1 for x.

Multiply both sides of the equation by 3.

x+2<1⋅(3)

Remove parentheses.

x+2<1⋅(3)

Multiply 3 by 1.

x+2<3

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

Subtract 2 from both sides of the inequality.

x<3-2

Subtract 2 from 3.

x<1

x<1

x<1

Find the intersection of x<1 and x≥-2.

-2≤x<1

-2≤x<1

Solve -x+23<1 for x.

Multiply each term in -x+23<1 by -1

Multiply each term in -x+23<1 by -1. When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

(-x+23)⋅-1>1⋅-1

Multiply (-x+23)⋅-1.

Multiply -1 by -1.

1x+23>1⋅-1

Multiply x+23 by 1.

x+23>1⋅-1

x+23>1⋅-1

Multiply -1 by 1.

x+23>-1

x+23>-1

Multiply both sides of the equation by 3.

x+2>-1⋅3

Remove parentheses.

x+2>-1⋅3

Multiply -1 by 3.

x+2>-3

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

Subtract 2 from both sides of the inequality.

x>-3-2

Subtract 2 from -3.

x>-5

x>-5

x>-5

Find the intersection of x>-5 and x<-2.

-5<x<-2

-5<x<-2

Find the union of the solutions.

-5<x<1

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