Solve 2x-1/3x+4<5

$$\frac{ 2x-1 }{ 3x+4 } < 5$$

Solve for x

$$x\in \left(-\infty,-\frac{21}{13}\right)\cup \left(-\frac{4}{3},\infty\right)$$

Solution Steps

Denominator 3x+4 cannot be zero since division by zero is not defined. There are two cases.

$$3x+4>0 3x+4<0$$

Consider the case when 3x+4 is positive. Move 4 to the right hand side.

$$3x>-4$$

Divide both sides by 3. Since 3 is positive, the inequality direction remains the same.

$$x>-\frac{4}{3}$$

The initial inequality does not change the direction when multiplied by 3x+4 for 3x+4>0.

$$2x-1<5\left(3x+4\right)$$

Multiply out the right hand side.

$$2x-1<15x+20$$

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

$$2x-15x<1+20$$

Combine like terms.

$$-13x<21$$

Divide both sides by -13. Since -13 is negative, the inequality direction is changed.

$$x>-\frac{21}{13}$$

Consider condition x>-\frac{4}{3} specified above.

$$x>-\frac{4}{3}$$

Now consider the case when 3x+4 is negative. Move 4 to the right hand side.

$$3x<-4$$

Divide both sides by 3. Since 3 is positive, the inequality direction remains the same.

$$x<-\frac{4}{3}$$

The initial inequality changes the direction when multiplied by 3x+4 for 3x+4<0.

$$2x-1>5\left(3x+4\right)$$

Multiply out the right hand side.

$$2x-1>15x+20$$

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

$$2x-15x>1+20$$

Combine like terms.

$$-13x>21$$

Divide both sides by -13. Since -13 is negative, the inequality direction is changed.

$$x<-\frac{21}{13}$$

Consider condition x<-\frac{4}{3} specified above. The result remains the same.

$$x<-\frac{21}{13}$$

The final solution is the union of the obtained solutions.

$$x\in \left(-\infty,-\frac{21}{13}\right)\cup \left(-\frac{4}{3},\infty\right)$$