# Solve for x |3+4x|-4>3 |3+4x|-4>3
Write |3+4x|-4>3 as a piecewise.
To find the interval for the first piece, find where the inside of the absolute value is non-negative.
3+4x≥0
Solve the inequality.
Subtract 3 from both sides of the inequality.
4x≥-3
Divide each term by 4 and simplify.
Divide each term in 4x≥-3 by 4.
4×4≥-34
Cancel the common factor of 4.
Cancel the common factor.
4×4≥-34
Divide x by 1.
x≥-34
x≥-34
Move the negative in front of the fraction.
x≥-34
x≥-34
x≥-34
In the piece where 3+4x is non-negative, remove the absolute value.
3+4x-4>3
To find the interval for the second piece, find where the inside of the absolute value is negative.
3+4x<0
Solve the inequality.
Subtract 3 from both sides of the inequality.
4x<-3
Divide each term by 4 and simplify.
Divide each term in 4x<-3 by 4.
4×4<-34
Cancel the common factor of 4.
Cancel the common factor.
4×4<-34
Divide x by 1.
x<-34
x<-34
Move the negative in front of the fraction.
x<-34
x<-34
x<-34
In the piece where 3+4x is negative, remove the absolute value and multiply by -1.
-(3+4x)-4>3
Write as a piecewise.
{3+4x-4>3x≥-34-(3+4x)-4>3x<-34
Subtract 4 from 3.
{4x-1>3x≥-34-(3+4x)-4>3x<-34
Simplify -(3+4x)-4>3.
Simplify each term.
Apply the distributive property.
{4x-1>3x≥-34-1⋅3-(4x)-4>3x<-34
Multiply -1 by 3.
{4x-1>3x≥-34-3-(4x)-4>3x<-34
Multiply 4 by -1.
{4x-1>3x≥-34-3-4x-4>3x<-34
{4x-1>3x≥-34-3-4x-4>3x<-34
Subtract 4 from -3.
{4x-1>3x≥-34-4x-7>3x<-34
{4x-1>3x≥-34-4x-7>3x<-34
{4x-1>3x≥-34-4x-7>3x<-34
Solve 4x-1>3 when x≥-34.
Solve 4x-1>3 for x.
Move all terms not containing x to the right side of the inequality.
Add 1 to both sides of the inequality.
4x>3+1
Add 3 and 1.
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≥-34.
x>1
x>1
Solve -4x-7>3 when x<-34.
Solve -4x-7>3 for x.
Move all terms not containing x to the right side of the inequality.
Add 7 to both sides of the inequality.
-4x>3+7
Add 3 and 7.
-4x>10
-4x>10
Divide each term by -4 and simplify.
Divide each term in -4x>10 by -4. When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.
-4x-4<10-4
Cancel the common factor of -4.
Cancel the common factor.
-4x-4<10-4
Divide x by 1.
x<10-4
x<10-4
Simplify 10-4.
Cancel the common factor of 10 and -4.
Factor 2 out of 10.
x<2(5)-4
Cancel the common factors.
Factor 2 out of -4.
x<2⋅52⋅-2
Cancel the common factor.
x<2⋅52⋅-2
Rewrite the expression.
x<5-2
x<5-2
x<5-2
Move the negative in front of the fraction.
x<-52
x<-52
x<-52
x<-52
Find the intersection of x<-52 and x<-34.
x<-52
x<-52
Find the union of the solutions.
x<-52 or x>1
The result can be shown in multiple forms.
Inequality Form:
x<-52 or x>1
Interval Notation:
(-∞,-52)∪(1,∞)
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