|3x-1|>5

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

3x-1≥0

Solve the inequality.

Add 1 to both sides of the inequality.

3x≥1

Divide each term by 3 and simplify.

Divide each term in 3x≥1 by 3.

3×3≥13

Cancel the common factor of 3.

Cancel the common factor.

3×3≥13

Divide x by 1.

x≥13

x≥13

x≥13

x≥13

In the piece where 3x-1 is non-negative, remove the absolute value.

3x-1>5

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

3x-1<0

Solve the inequality.

Add 1 to both sides of the inequality.

3x<1

Divide each term by 3 and simplify.

Divide each term in 3x<1 by 3.

3×3<13

Cancel the common factor of 3.

Cancel the common factor.

3×3<13

Divide x by 1.

x<13

x<13

x<13

x<13

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

-(3x-1)>5

Write as a piecewise.

{3x-1>5x≥13-(3x-1)>5x<13

Simplify -(3x-1)>5.

Apply the distributive property.

{3x-1>5x≥13-(3x)–1>5x<13

Multiply 3 by -1.

{3x-1>5x≥13-3x–1>5x<13

Multiply -1 by -1.

{3x-1>5x≥13-3x+1>5x<13

{3x-1>5x≥13-3x+1>5x<13

{3x-1>5x≥13-3x+1>5x<13

Solve 3x-1>5 for x.

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

Add 1 to both sides of the inequality.

3x>5+1

Add 5 and 1.

3x>6

3x>6

Divide each term by 3 and simplify.

Divide each term in 3x>6 by 3.

3×3>63

Cancel the common factor of 3.

Cancel the common factor.

3×3>63

Divide x by 1.

x>63

x>63

Divide 6 by 3.

x>2

x>2

x>2

Find the intersection of x>2 and x≥13.

x>2

x>2

Solve -3x+1>5 for x.

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

Subtract 1 from both sides of the inequality.

-3x>5-1

Subtract 1 from 5.

-3x>4

-3x>4

Divide each term by -3 and simplify.

Divide each term in -3x>4 by -3. When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

-3x-3<4-3

Cancel the common factor of -3.

Cancel the common factor.

-3x-3<4-3

Divide x by 1.

x<4-3

x<4-3

Move the negative in front of the fraction.

x<-43

x<-43

x<-43

Find the intersection of x<-43 and x<13.

x<-43

x<-43

Find the union of the solutions.

x<-43 or x>2

The result can be shown in multiple forms.

Inequality Form:

x<-43 or x>2

Interval Notation:

(-∞,-43)∪(2,∞)

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