|2x+7|>27

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.

Subtract 7 from 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

Move the negative in front of the fraction.

x≥-72

x≥-72

x≥-72

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

2x+7>27

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

2x+7<0

Solve the inequality.

Subtract 7 from 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

Move the negative in front of the fraction.

x<-72

x<-72

x<-72

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

-(2x+7)>27

Write as a piecewise.

{2x+7>27x≥-72-(2x+7)>27x<-72

Simplify -(2x+7)>27.

Apply the distributive property.

{2x+7>27x≥-72-(2x)-1⋅7>27x<-72

Multiply 2 by -1.

{2x+7>27x≥-72-2x-1⋅7>27x<-72

Multiply -1 by 7.

{2x+7>27x≥-72-2x-7>27x<-72

{2x+7>27x≥-72-2x-7>27x<-72

{2x+7>27x≥-72-2x-7>27x<-72

Solve 2x+7>27 for x.

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

Subtract 7 from both sides of the inequality.

2x>27-7

Subtract 7 from 27.

2x>20

2x>20

Divide each term by 2 and simplify.

Divide each term in 2x>20 by 2.

2×2>202

Cancel the common factor of 2.

Cancel the common factor.

2×2>202

Divide x by 1.

x>202

x>202

Divide 20 by 2.

x>10

x>10

x>10

Find the intersection of x>10 and x≥-72.

x>10

x>10

Solve -2x-7>27 for x.

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

Add 7 to both sides of the inequality.

-2x>27+7

Add 27 and 7.

-2x>34

-2x>34

Divide each term by -2 and simplify.

Divide each term in -2x>34 by -2. When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

-2x-2<34-2

Cancel the common factor of -2.

Cancel the common factor.

-2x-2<34-2

Divide x by 1.

x<34-2

x<34-2

Divide 34 by -2.

x<-17

x<-17

x<-17

Find the intersection of x<-17 and x<-72.

x<-17

x<-17

Find the union of the solutions.

x<-17 or x>10

The result can be shown in multiple forms.

Inequality Form:

x<-17 or x>10

Interval Notation:

(-∞,-17)∪(10,∞)

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