Solve 3x-7/4x+2<7/15

$$\frac { 3 x - 7 } { 4 x + 2 } < \frac { 7 } { 15 }$$

Solve for x

$$x\in \left(-\frac{1}{2},7\right)$$

Solution Steps

Denominator 4x+2 cannot be zero since division by zero is not defined. There are two cases.

$$4x+2>0 4x+2<0$$

Consider the case when 4x+2 is positive. Move 2 to the right hand side.

$$4x>-2$$

Divide both sides by 4. Since 4 is positive, the inequality direction remains the same.

$$x>-\frac{1}{2}$$

The initial inequality does not change the direction when multiplied by 4x+2 for 4x+2>0.

$$3x-7<\frac{7}{15}\left(4x+2\right)$$

Multiply out the right hand side.

$$3x-7<\frac{28}{15}x+\frac{14}{15}$$

Move the terms containing x to the left hand side and all other terms to the right hand side.

$$3x-\frac{28}{15}x<7+\frac{14}{15}$$

Combine like terms.

$$\frac{17}{15}x<\frac{119}{15}$$

Divide both sides by \frac{17}{15}. Since \frac{17}{15} is positive, the inequality direction remains the same.

$$x<7$$

Consider condition x>-\frac{1}{2} specified above.

$$x\in \left(-\frac{1}{2},7\right)$$

Now consider the case when 4x+2 is negative. Move 2 to the right hand side.

$$4x<-2$$

Divide both sides by 4. Since 4 is positive, the inequality direction remains the same.

$$x<-\frac{1}{2}$$

The initial inequality changes the direction when multiplied by 4x+2 for 4x+2<0.

$$3x-7>\frac{7}{15}\left(4x+2\right)$$

Multiply out the right hand side.

$$3x-7>\frac{28}{15}x+\frac{14}{15}$$

Move the terms containing x to the left hand side and all other terms to the right hand side.

$$3x-\frac{28}{15}x>7+\frac{14}{15}$$

Combine like terms.

$$\frac{17}{15}x>\frac{119}{15}$$

Divide both sides by \frac{17}{15}. Since \frac{17}{15} is positive, the inequality direction remains the same.

$$x>7$$

Consider condition x<-\frac{1}{2} specified above.

$$x\in \emptyset$$

The final solution is the union of the obtained solutions.

$$x\in \left(-\frac{1}{2},7\right)$$