|4x+2|+3>9

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

4x+2≥0

Solve the inequality.

Subtract 2 from both sides of the inequality.

4x≥-2

Divide each term by 4 and simplify.

Divide each term in 4x≥-2 by 4.

4×4≥-24

Cancel the common factor of 4.

Cancel the common factor.

4×4≥-24

Divide x by 1.

x≥-24

x≥-24

Simplify -24.

Cancel the common factor of -2 and 4.

Factor 2 out of -2.

x≥2(-1)4

Cancel the common factors.

Factor 2 out of 4.

x≥2⋅-12⋅2

Cancel the common factor.

x≥2⋅-12⋅2

Rewrite the expression.

x≥-12

x≥-12

x≥-12

Move the negative in front of the fraction.

x≥-12

x≥-12

x≥-12

x≥-12

In the piece where 4x+2 is non-negative, remove the absolute value.

4x+2+3>9

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

4x+2<0

Solve the inequality.

Subtract 2 from both sides of the inequality.

4x<-2

Divide each term by 4 and simplify.

Divide each term in 4x<-2 by 4.

4×4<-24

Cancel the common factor of 4.

Cancel the common factor.

4×4<-24

Divide x by 1.

x<-24

x<-24

Simplify -24.

Cancel the common factor of -2 and 4.

Factor 2 out of -2.

x<2(-1)4

Cancel the common factors.

Factor 2 out of 4.

x<2⋅-12⋅2

Cancel the common factor.

x<2⋅-12⋅2

Rewrite the expression.

x<-12

x<-12

x<-12

Move the negative in front of the fraction.

x<-12

x<-12

x<-12

x<-12

In the piece where 4x+2 is negative, remove the absolute value and multiply by -1.

-(4x+2)+3>9

Write as a piecewise.

{4x+2+3>9x≥-12-(4x+2)+3>9x<-12

Add 2 and 3.

{4x+5>9x≥-12-(4x+2)+3>9x<-12

Simplify -(4x+2)+3>9.

Simplify each term.

Apply the distributive property.

{4x+5>9x≥-12-(4x)-1⋅2+3>9x<-12

Multiply 4 by -1.

{4x+5>9x≥-12-4x-1⋅2+3>9x<-12

Multiply -1 by 2.

{4x+5>9x≥-12-4x-2+3>9x<-12

{4x+5>9x≥-12-4x-2+3>9x<-12

Add -2 and 3.

{4x+5>9x≥-12-4x+1>9x<-12

{4x+5>9x≥-12-4x+1>9x<-12

{4x+5>9x≥-12-4x+1>9x<-12

Solve 4x+5>9 for x.

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

Subtract 5 from both sides of the inequality.

4x>9-5

Subtract 5 from 9.

4x>4

4x>4

Divide each term by 4 and simplify.

Divide each term in 4x>4 by 4.

4×4>44

Cancel the common factor of 4.

Cancel the common factor.

4×4>44

Divide x by 1.

x>44

x>44

Divide 4 by 4.

x>1

x>1

x>1

Find the intersection of x>1 and x≥-12.

x>1

x>1

Solve -4x+1>9 for x.

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

Subtract 1 from both sides of the inequality.

-4x>9-1

Subtract 1 from 9.

-4x>8

-4x>8

Divide each term by -4 and simplify.

Divide each term in -4x>8 by -4. When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

-4x-4<8-4

Cancel the common factor of -4.

Cancel the common factor.

-4x-4<8-4

Divide x by 1.

x<8-4

x<8-4

Divide 8 by -4.

x<-2

x<-2

x<-2

Find the intersection of x<-2 and x<-12.

x<-2

x<-2

Find the union of the solutions.

x<-2 or x>1

The result can be shown in multiple forms.

Inequality Form:

x<-2 or x>1

Interval Notation:

(-∞,-2)∪(1,∞)

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