|x-2|<5

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

x-2≥0

Add 2 to both sides of the inequality.

x≥2

In the piece where x-2 is non-negative, remove the absolute value.

x-2<5

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

x-2<0

Add 2 to both sides of the inequality.

x<2

In the piece where x-2 is negative, remove the absolute value and multiply by -1.

-(x-2)<5

Write as a piecewise.

{x-2<5x≥2-(x-2)<5x<2

Simplify -(x-2)<5.

Apply the distributive property.

{x-2<5x≥2-x–2<5x<2

Multiply -1 by -2.

{x-2<5x≥2-x+2<5x<2

{x-2<5x≥2-x+2<5x<2

{x-2<5x≥2-x+2<5x<2

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

Add 2 to both sides of the inequality.

x<5+2

Add 5 and 2.

x<7

x<7

Find the intersection of x<7 and x≥2.

2≤x<7

2≤x<7

Solve -x+2<5 for x.

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

Subtract 2 from both sides of the inequality.

-x<5-2

Subtract 2 from 5.

-x<3

-x<3

Multiply each term in -x<3 by -1

Multiply each term in -x<3 by -1. When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

(-x)⋅-1>3⋅-1

Multiply (-x)⋅-1.

Multiply -1 by -1.

1x>3⋅-1

Multiply x by 1.

x>3⋅-1

x>3⋅-1

Multiply 3 by -1.

x>-3

x>-3

x>-3

Find the intersection of x>-3 and x<2.

-3<x<2

-3<x<2

Find the union of the solutions.

-3<x<7

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