Question

Решить неравенство ((x+4) * |x-18|)^2 < (10x + 19) * (x-18)^2

Answer 1

((x+4) * |x-18|)^2 < (10x + 19) * (x-18)^2

(x+4)^2 * (x-18)^2 < (10x + 19) * (x-18)^2

(x+4)^2 < 10x + 19

x^2 + 8x + 16 - 10x - 19 < 0

x^2 - 2x - 3 < 0 - решаем квадратное уравнение

x1=-1; x2=3;

(x+1)(x-3) < 0

x ∈ (-1; 3)

Answer 2

To solve the inequality $((x+4) \cdot |x-18|)^2 < (10x + 19) \cdot (x-18)^2$, we need to follow these steps:

Step 1: Identify the critical points (values of x that make the expressions inside the absolute value and square zero). Step 2: Determine the sign of the expressions within the absolute value for different intervals. Step 3: Solve the inequality for each interval. Step 4: Combine the solutions to obtain the final solution.

Let's start solving the inequality step-by-step:

Step 1: Identify critical points. The expressions inside the absolute value and square are zero when:

1. $x+4 = 0 \implies x = -4$.
2. $|x-18| = 0 \implies x = 18$.
3. $x-18 = 0 \implies x = 18$ (But this is already covered by the second point).

Step 2: Determine the sign of expressions in different intervals. We'll consider three intervals based on the critical points: $(- \infty, -4)$, $(-4, 18)$, and $(18, +\infty)$.

In $(- \infty, -4)$:

• $x + 4 < 0$
• $|x - 18| > 0$
• $10x + 19 < 0$
• $x - 18 < 0$

In $(-4, 18)$:

• $x + 4 > 0$
• $|x - 18| > 0$
• $10x + 19 > 0$
• $x - 18 < 0$

In $(18, +\infty)$:

• $x + 4 > 0$
• $|x - 18| > 0$
• $10x + 19 > 0$
• $x - 18 > 0$

Step 3: Solve the inequality for each interval.

For $(- \infty, -4)$: $(x + 4)^2 \cdot (18 - x)^2 < (10x + 19) \cdot (x - 18)^2$ This will simplify to $100x^2 - 172x + 19 > 0$. Solving this quadratic inequality, we get the solution: $-\frac{19}{50} < x < \frac{9}{5}$.

For $(-4, 18)$: $(x + 4)^2 \cdot (18 - x)^2 < (10x + 19) \cdot (x - 18)^2$ This will simplify to $64x^2 - 292x + 361 > 0$. Solving this quadratic inequality, we get the solution: $-\frac{19}{8} < x < 18$.

For $(18, +\infty)$: $(x + 4)^2 \cdot (x - 18)^2 < (10x + 19) \cdot (x - 18)^2$ Since $|x - 18| > 0$ in this interval, we can divide both sides by $(x - 18)^2$ (without flipping the inequality) to get: $(x + 4)^2 < 10x + 19$. Expanding and rearranging, we get $x^2 - 2x - 3 < 0$. Solving this quadratic inequality, we get the solution: $x < -1$ or $x > 3$.

Step 4: Combine the solutions. Finally, combining the solutions from each interval, we have the solution to the inequality: $-\frac{19}{50} < x < -1$ or $\frac{9}{5} < x < 3$.

Therefore, the solution to the inequality $((x+4) \cdot |x-18|)^2 < (10x + 19) \cdot (x-18)^2$