Solve -7x+3/2x+8>4

$$\frac { - 7 x + 3 } { 2 x + 8 } > 4$$

Solve for x

$$x\in \left(-4,-\frac{29}{15}\right)$$

Solution Steps

Denominator 2x+8 cannot be zero since division by zero is not defined. There are two cases.

$$2x+8>0 2x+8<0$$

Consider the case when 2x+8 is positive. Move 8 to the right hand side.

$$2x>-8$$

Divide both sides by 2. Since 2 is positive, the inequality direction remains the same.

$$x>-4$$

The initial inequality does not change the direction when multiplied by 2x+8 for 2x+8>0.

$$-7x+3>4\left(2x+8\right)$$

Multiply out the right hand side.

$$-7x+3>8x+32$$

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

$$-7x-8x>-3+32$$

Combine like terms.

$$-15x>29$$

Divide both sides by -15. Since -15 is negative, the inequality direction is changed.

$$x<-\frac{29}{15}$$

Consider condition x>-4 specified above.

$$x\in \left(-4,-\frac{29}{15}\right)$$

Now consider the case when 2x+8 is negative. Move 8 to the right hand side.

$$2x<-8$$

Divide both sides by 2. Since 2 is positive, the inequality direction remains the same.

$$x<-4$$

The initial inequality changes the direction when multiplied by 2x+8 for 2x+8<0.

$$-7x+3<4\left(2x+8\right)$$

Multiply out the right hand side.

$$-7x+3<8x+32$$

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

$$-7x-8x<-3+32$$

Combine like terms.

$$-15x<29$$

Divide both sides by -15. Since -15 is negative, the inequality direction is changed.

$$x>-\frac{29}{15}$$

Consider condition x<-4 specified above.

$$x\in \emptyset$$

The final solution is the union of the obtained solutions.

$$x\in \left(-4,-\frac{29}{15}\right)$$