3|3-x|+1<5

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

3-x≥0

Solve the inequality.

Subtract 3 from both sides of the inequality.

-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

In the piece where 3-x is non-negative, remove the absolute value.

3(3-x)+1<5

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

3-x<0

Solve the inequality.

Subtract 3 from both sides of the inequality.

-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

In the piece where 3-x is negative, remove the absolute value and multiply by -1.

3(-(3-x))+1<5

Write as a piecewise.

{3(3-x)+1<5x≤33(-(3-x))+1<5x>3

Simplify 3(3-x)+1<5.

Simplify each term.

Apply the distributive property.

{3⋅3+3(-x)+1<5x≤33(-(3-x))+1<5x>3

Multiply 3 by 3.

{9+3(-x)+1<5x≤33(-(3-x))+1<5x>3

Multiply -1 by 3.

{9-3x+1<5x≤33(-(3-x))+1<5x>3

{9-3x+1<5x≤33(-(3-x))+1<5x>3

Add 9 and 1.

{-3x+10<5x≤33(-(3-x))+1<5x>3

{-3x+10<5x≤33(-(3-x))+1<5x>3

Simplify 3(-(3-x))+1<5.

Simplify each term.

Apply the distributive property.

{-3x+10<5x≤33(-1⋅3–x)+1<5x>3

Multiply -1 by 3.

{-3x+10<5x≤33(-3–x)+1<5x>3

Multiply –x.

Multiply -1 by -1.

{-3x+10<5x≤33(-3+1x)+1<5x>3

Multiply x by 1.

{-3x+10<5x≤33(-3+x)+1<5x>3

{-3x+10<5x≤33(-3+x)+1<5x>3

Apply the distributive property.

{-3x+10<5x≤33⋅-3+3x+1<5x>3

Multiply 3 by -3.

{-3x+10<5x≤3-9+3x+1<5x>3

{-3x+10<5x≤3-9+3x+1<5x>3

Add -9 and 1.

{-3x+10<5x≤33x-8<5x>3

{-3x+10<5x≤33x-8<5x>3

{-3x+10<5x≤33x-8<5x>3

Solve -3x+10<5 for x.

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

Subtract 10 from both sides of the inequality.

-3x<5-10

Subtract 10 from 5.

-3x<-5

-3x<-5

Divide each term by -3 and simplify.

Divide each term in -3x<-5 by -3. When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

-3x-3>-5-3

Cancel the common factor of -3.

Cancel the common factor.

-3x-3>-5-3

Divide x by 1.

x>-5-3

x>-5-3

Dividing two negative values results in a positive value.

x>53

x>53

x>53

Find the intersection of x>53 and x≤3.

53<x≤3

53<x≤3

Solve 3x-8<5 for x.

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

Add 8 to both sides of the inequality.

3x<5+8

Add 5 and 8.

3x<13

3x<13

Divide each term by 3 and simplify.

Divide each term in 3x<13 by 3.

3×3<133

Cancel the common factor of 3.

Cancel the common factor.

3×3<133

Divide x by 1.

x<133

x<133

x<133

x<133

Find the intersection of x<133 and x>3.

3<x<133

3<x<133

Find the union of the solutions.

53<x<133

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