|3+4x|-4>3

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 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 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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