Solve -7-x/-7+xleq1

$$\frac { - 7 - x } { - 7 + x } \leq 1$$

Solve for x

$$x\in (-\infty,0]\cup (7,\infty)$$

Solution Steps

Denominator -7+x cannot be zero since division by zero is not defined. There are two cases.

$$-7+x>0 -7+x<0$$

Consider the case when -7+x is positive. Move -7 to the right hand side.

$$x>7$$

The initial inequality does not change the direction when multiplied by -7+x for -7+x>0.

$$-7-x\leq -7+x$$

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

$$-x-x\leq 7-7$$

Combine like terms.

$$-2x\leq 0$$

Divide both sides by -2. Since -2 is negative, the inequality direction is changed.

$$x\geq 0$$

Consider condition x>7 specified above.

$$x>7$$

Now consider the case when -7+x is negative. Move -7 to the right hand side.

$$x<7$$

The initial inequality changes the direction when multiplied by -7+x for -7+x<0.

$$-7-x\geq -7+x$$

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

$$-x-x\geq 7-7$$

Combine like terms.

$$-2x\geq 0$$

Divide both sides by -2. Since -2 is negative, the inequality direction is changed.

$$x\leq 0$$

The final solution is the union of the obtained solutions.

$$x\in (-\infty,0]\cup (7,\infty)$$