Solve x-2/3x+5leq4

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

Solve for x

$$x\in (-\infty,-2]\cup (-\frac{5}{3},\infty)$$

Solution Steps

Denominator 3x+5 cannot be zero since division by zero is not defined. There are two cases.

$$3x+5>0 3x+5<0$$

Consider the case when 3x+5 is positive. Move 5 to the right hand side.

$$3x>-5$$

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

$$x>-\frac{5}{3}$$

The initial inequality does not change the direction when multiplied by 3x+5 for 3x+5>0.

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

Multiply out the right hand side.

$$x-2\leq 12x+20$$

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

$$x-12x\leq 2+20$$

Combine like terms.

$$-11x\leq 22$$

Divide both sides by -11. Since -11 is negative, the inequality direction is changed.

$$x\geq -2$$

Consider condition x>-\frac{5}{3} specified above.

$$x>-\frac{5}{3}$$

Now consider the case when 3x+5 is negative. Move 5 to the right hand side.

$$3x<-5$$

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

$$x<-\frac{5}{3}$$

The initial inequality changes the direction when multiplied by 3x+5 for 3x+5<0.

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

Multiply out the right hand side.

$$x-2\geq 12x+20$$

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

$$x-12x\geq 2+20$$

Combine like terms.

$$-11x\geq 22$$

Divide both sides by -11. Since -11 is negative, the inequality direction is changed.

$$x\leq -2$$

The final solution is the union of the obtained solutions.

$$x\in (-\infty,-2]\cup (-\frac{5}{3},\infty)$$