Question

Неравенство: x-4+√x-4/x-2<6/x-2

Answer 1

X-4+√x-4/x-2<6/x-2
x-4+√x-4/x-2 - 6/x-2<0
x-4+√x-4 - 6/x-2<0
x-10+√x-4/x-2<0
х-2>0
х>2
x-10+√x-4<0
x-4<(х-10)^2
х-4<х^2-20х+100
-х^2+х+20х<4+100
-х^2+21х-104<0
х^2-21х+104>0
х1=8
х2=13
(х-8)(х-13)>0
х>8 х<8
х>13 х<13
х>2

Answer 2

To solve the inequality $x-4+\frac{\sqrt{x-4}}{x-2} < \frac{6}{x-2}$, we need to find the range of values for $x$ that satisfy the inequality. Here's how we can do it step by step:

Step 1: Get rid of the fraction on the left side of the inequality. Multiply both sides of the inequality by $x-2$ to eliminate the fraction: $(x-2)(x-4) + \sqrt{x-4} < 6$.

Step 2: Handle the square root term. To make it easier, let's make a substitution: Let $y = \sqrt{x-4}$. Now the inequality becomes: $(x-2)(x-4) + y < 6$.

Step 3: Solve for $y$. Rearrange the inequality: $y < 6 - (x-2)(x-4)$.

Step 4: Solve for $x$ in terms of $y$. To do this, we need to find the values of $x$ that make the quadratic expression $6 - (x-2)(x-4)$ non-negative (i.e., $6 - (x-2)(x-4) \geq 0$), since the square root of a non-negative number is always real.

$6 - (x-2)(x-4) \geq 0$.

Step 5: Factorize the quadratic expression. $6 - (x-2)(x-4) \geq 0$ can be simplified as: $6 - (x^2 - 6x + 8) \geq 0$.

Step 6: Combine like terms. $6 - x^2 + 6x - 8 \geq 0$.

Step 7: Simplify further. $-x^2 + 6x - 2 \geq 0$.

Step 8: Solve the quadratic inequality. To find the critical points, set the expression equal to zero: $-x^2 + 6x - 2 = 0$.

Now, you can use the quadratic formula to solve for $x$: $x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$.

For our equation, $a = -1$, $b = 6$, and $c = -2$: $x = \frac{-6 \pm \sqrt{6^2-4(-1)(-2)}}{2(-1)}$.

$x = \frac{-6 \pm \sqrt{36-8}}{-2}$.

$x = \frac{-6 \pm \sqrt{28}}{-2}$.

$x = \frac{-6 \pm 2\sqrt{7}}{-2}$.

Now, we have two solutions for $x$: $x = 3 + \sqrt{7}$ and $x = 3 - \sqrt{7}$.

Step 9: Determine the sign of the quadratic expression in the interval $(3 - \sqrt{7}), (3 + \sqrt{7})$. We know that the coefficient of $x^2$ is negative ($-1$), so the parabola opens downward. The expression is positive in the interval between the two critical points and negative outside this interval.

Step 10: Final solution. Since the expression $6 - (x-2)(x-4)$