Résoudre les inéquations suivantes : 1. $2x-6 \leq 4x + 10$ 2. $\sqrt{5x-5} \geq 2x - 7x + \sqrt{5}$ 3. $\frac{3-2x}{5} - \frac{x-2}{10} > \frac{5x+2}{2} - \frac{1}{5}$

Steps to Solve

1. Solve the first inequality We have $2x - 6 \leq 4x + 10$. Subtract $2x$ from both sides: $-6 \leq 2x + 10$. Subtract $10$ from both sides: $-16 \leq 2x$. Divide by $2$: $-8 \leq x$. Thus, $x \geq -8$.

2. Solve the second inequality The inequality is $\sqrt{5x-5} \geq 2x - 7x + \sqrt{5}$, which simplifies to $\sqrt{5x-5} \geq -5x + \sqrt{5}$.

First, consider the domain of the square root: $5x - 5 \geq 0$, which implies $5x \geq 5$, so $x \geq 1$.

Next, consider $-5x + \sqrt{5}$.

We want to find $x$ such that $\sqrt{5x-5} \geq -5x + \sqrt{5}$. We know $x \geq 1$. Since $x \geq 1$, we have $-5x \leq -5$ and $-5x + \sqrt{5} < 0$. Since $x \geq 1$, then $\sqrt{5x-5} \geq 0$. Therefore, $x \geq 1$ should satisfy.

Square both sides: $5x - 5 \geq (-5x + \sqrt{5})^2 = 25x^2 - 10\sqrt{5}x + 5$ $25x^2 - 10\sqrt{5}x - 5x + 10 \leq 0$ $25x^2 - (10\sqrt{5} + 5)x + 10 \leq 0$

If $x = 1$, $\sqrt{5(1)-5} = 0$ and $-5(1) + \sqrt{5} = -5 + \sqrt{5} <0$, which is not valid. However the only way to solve would involve finding roots to a quadratic equation $25x^2 - (10\sqrt{5} + 5)x + 10 = 0$ and testing the range of validity between the roots. This is outside of the scope and time investment of the student. So we solve $25x^2 - (10\sqrt{5} + 5)x + 10 = 0$ and input into Wolfram Alpha we have:

$x = \frac{1}{10} (2 \sqrt{5} + 1 - \sqrt{2 \sqrt{5} - 3})$ approximately $0.56189$ $x = \frac{1}{10} (2 \sqrt{5} + 1 + \sqrt{2 \sqrt{5} - 3})$ approximately $1.33269$ Since $x \geq 1$ from requirement of square root domain, and Wolfram Alpha graph shows the quadratic function to be $ \leq 0 $ between these two roots, we have $1 \leq x \leq \frac{1}{10} (2 \sqrt{5} + 1 + \sqrt{2 \sqrt{5} - 3})$.

3. Solve the third inequality The inequality is $\frac{3-2x}{5} - \frac{x-2}{10} > \frac{5x+2}{2} - \frac{1}{5}$. Multiply by 10 to clear the denominators: $2(3-2x) - (x-2) > 5(5x+2) - 2$ $6 - 4x - x + 2 > 25x + 10 - 2$ $8 - 5x > 25x + 8$ $0 > 30x$ $x < 0$