|3x-4|>5

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

3x-4≥0

Solve the inequality.

Add 4 to both sides of the inequality.

3x≥4

Divide each term by 3 and simplify.

Divide each term in 3x≥4 by 3.

3×3≥43

Cancel the common factor of 3.

Cancel the common factor.

3×3≥43

Divide x by 1.

x≥43

x≥43

x≥43

x≥43

In the piece where 3x-4 is non-negative, remove the absolute value.

3x-4>5

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

3x-4<0

Solve the inequality.

Add 4 to both sides of the inequality.

3x<4

Divide each term by 3 and simplify.

Divide each term in 3x<4 by 3.

3×3<43

Cancel the common factor of 3.

Cancel the common factor.

3×3<43

Divide x by 1.

x<43

x<43

x<43

x<43

In the piece where 3x-4 is negative, remove the absolute value and multiply by -1.

-(3x-4)>5

Write as a piecewise.

{3x-4>5x≥43-(3x-4)>5x<43

Simplify -(3x-4)>5.

Apply the distributive property.

{3x-4>5x≥43-(3x)–4>5x<43

Multiply 3 by -1.

{3x-4>5x≥43-3x–4>5x<43

Multiply -1 by -4.

{3x-4>5x≥43-3x+4>5x<43

{3x-4>5x≥43-3x+4>5x<43

{3x-4>5x≥43-3x+4>5x<43

Solve 3x-4>5 for x.

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

Add 4 to both sides of the inequality.

3x>5+4

Add 5 and 4.

3x>9

3x>9

Divide each term by 3 and simplify.

Divide each term in 3x>9 by 3.

3×3>93

Cancel the common factor of 3.

Cancel the common factor.

3×3>93

Divide x by 1.

x>93

x>93

Divide 9 by 3.

x>3

x>3

x>3

Find the intersection of x>3 and x≥43.

x>3

x>3

Solve -3x+4>5 for x.

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

Subtract 4 from both sides of the inequality.

-3x>5-4

Subtract 4 from 5.

-3x>1

-3x>1

Divide each term by -3 and simplify.

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

-3x-3<1-3

Cancel the common factor of -3.

Cancel the common factor.

-3x-3<1-3

Divide x by 1.

x<1-3

x<1-3

Move the negative in front of the fraction.

x<-13

x<-13

x<-13

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

x<-13

x<-13

Find the union of the solutions.

x<-13 or x>3

The result can be shown in multiple forms.

Inequality Form:

x<-13 or x>3

Interval Notation:

(-∞,-13)∪(3,∞)

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