Solve x+1/2x-4geq-1

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

Solve for x

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

Solution Steps

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

$$2x-4>0 2x-4<0$$

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

$$2x>4$$

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

$$x>2$$

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

$$x+1\geq -\left(2x-4\right)$$

Multiply out the right hand side.

$$x+1\geq -2x+4$$

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

$$x+2x\geq -1+4$$

Combine like terms.

$$3x\geq 3$$

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

$$x\geq 1$$

Consider condition x>2 specified above.

$$x>2$$

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

$$2x<4$$

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

$$x<2$$

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

$$x+1\leq -\left(2x-4\right)$$

Multiply out the right hand side.

$$x+1\leq -2x+4$$

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

$$x+2x\leq -1+4$$

Combine like terms.

$$3x\leq 3$$

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

$$x\leq 1$$

The final solution is the union of the obtained solutions.

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