Solve x-pi/x-4leqtan1

$$\frac { x - \pi } { x - 4 } \leq \tan 1$$

Solve for x

$$x\in [\frac{200000000000000000\pi -13964051942574068}{196508987014356483},4)$$

Solution Steps

Evaluate trigonometric functions in the problem

$$\frac{x - \pi}{x - 4} \leq 0.017455064928217585$$

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

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

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

$$x>4$$

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

$$x-\pi \leq 0.017455064928217585\left(x-4\right)$$

Multiply out the right hand side.

$$x-\pi \leq 0.017455064928217585x-0.06982025971287034$$

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

$$x-0.017455064928217585x\leq \pi -0.06982025971287034$$

Combine like terms.

$$0.982544935071782415x\leq \pi -0.06982025971287034$$

Divide both sides by 0.982544935071782415. Since 0.982544935071782415 is positive, the inequality direction remains the same.

$$x\leq \frac{200000000000000000\pi -13964051942574068}{196508987014356483}$$

Consider condition x>4 specified above.

$$x\in \emptyset$$

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

$$x<4$$

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

$$x-\pi \geq 0.017455064928217585\left(x-4\right)$$

Multiply out the right hand side.

$$x-\pi \geq 0.017455064928217585x-0.06982025971287034$$

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

$$x-0.017455064928217585x\geq \pi -0.06982025971287034$$

Combine like terms.

$$0.982544935071782415x\geq \pi -0.06982025971287034$$

Divide both sides by 0.982544935071782415. Since 0.982544935071782415 is positive, the inequality direction remains the same.

$$x\geq \frac{200000000000000000\pi -13964051942574068}{196508987014356483}$$

Consider condition x<4 specified above.

$$x\in [\frac{200000000000000000\pi -13964051942574068}{196508987014356483},4)$$

The final solution is the union of the obtained solutions.

$$x\in [\frac{200000000000000000\pi -13964051942574068}{196508987014356483},4)$$