# Solve for x |4x+2|+3>9 |4x+2|+3>9
Write |4x+2|+3>9 as a piecewise.
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
{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
{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 when x≥-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 when x<-12.
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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Solve for x |4x+2|+3>9     