Solve 7x-4/x+2geq1

$$\frac{ 7x-4 }{ x+2 } \geq 1$$

Solve for x

$$x\in (-\infty,-2)\cup [1,\infty)$$

Solution Steps

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

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

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

$$x>-2$$

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

$$7x-4\geq x+2$$

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

$$7x-x\geq 4+2$$

Combine like terms.

$$6x\geq 6$$

Divide both sides by 6. Since 6 is positive, the inequality direction remains the same.

$$x\geq 1$$

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

$$x<-2$$

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

$$7x-4\leq x+2$$

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

$$7x-x\leq 4+2$$

Combine like terms.

$$6x\leq 6$$

Divide both sides by 6. Since 6 is positive, the inequality direction remains the same.

$$x\leq 1$$

Consider condition x<-2 specified above.

$$x<-2$$

The final solution is the union of the obtained solutions.

$$x\in (-\infty,-2)\cup [1,\infty)$$