|2x-7|-1>0

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

2x-7≥0

Solve the inequality.

Add 7 to both sides of the inequality.

2x≥7

Divide each term by 2 and simplify.

Divide each term in 2x≥7 by 2.

2×2≥72

Cancel the common factor of 2.

Cancel the common factor.

2×2≥72

Divide x by 1.

x≥72

x≥72

x≥72

x≥72

In the piece where 2x-7 is non-negative, remove the absolute value.

2x-7-1>0

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

2x-7<0

Solve the inequality.

Add 7 to both sides of the inequality.

2x<7

Divide each term by 2 and simplify.

Divide each term in 2x<7 by 2.

2×2<72

Cancel the common factor of 2.

Cancel the common factor.

2×2<72

Divide x by 1.

x<72

x<72

x<72

x<72

In the piece where 2x-7 is negative, remove the absolute value and multiply by -1.

-(2x-7)-1>0

Write as a piecewise.

{2x-7-1>0x≥72-(2x-7)-1>0x<72

Subtract 1 from -7.

{2x-8>0x≥72-(2x-7)-1>0x<72

Simplify -(2x-7)-1>0.

Simplify each term.

Apply the distributive property.

{2x-8>0x≥72-(2x)–7-1>0x<72

Multiply 2 by -1.

{2x-8>0x≥72-2x–7-1>0x<72

Multiply -1 by -7.

{2x-8>0x≥72-2x+7-1>0x<72

{2x-8>0x≥72-2x+7-1>0x<72

Subtract 1 from 7.

{2x-8>0x≥72-2x+6>0x<72

{2x-8>0x≥72-2x+6>0x<72

{2x-8>0x≥72-2x+6>0x<72

Solve 2x-8>0 for x.

Add 8 to both sides of the inequality.

2x>8

Divide each term by 2 and simplify.

Divide each term in 2x>8 by 2.

2×2>82

Cancel the common factor of 2.

Cancel the common factor.

2×2>82

Divide x by 1.

x>82

x>82

Divide 8 by 2.

x>4

x>4

x>4

Find the intersection of x>4 and x≥72.

x>4

x>4

Solve -2x+6>0 for x.

Subtract 6 from both sides of the inequality.

-2x>-6

Divide each term by -2 and simplify.

Divide each term in -2x>-6 by -2. When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

-2x-2<-6-2

Cancel the common factor of -2.

Cancel the common factor.

-2x-2<-6-2

Divide x by 1.

x<-6-2

x<-6-2

Divide -6 by -2.

x<3

x<3

x<3

Find the intersection of x<3 and x<72.

x<3

x<3

Find the union of the solutions.

x<3 or x>4

The result can be shown in multiple forms.

Inequality Form:

x<3 or x>4

Interval Notation:

(-∞,3)∪(4,∞)

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