Question

5x²-20x≤0 x²≥16 x²+14x+49≤0

Answer 1

Ответ в файле

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Answer 2

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Answer 3

Let's solve each inequality step by step:

1. $5x^2 - 20x \leq 0$

First, factor out $5x$ from the expression:

$5x(x - 4) \leq 0$

Now, we have two factors: $5x$ and $x - 4$. To find the values of $x$ that satisfy the inequality, we can set each factor to zero and solve for $x$:

$5x = 0 \Rightarrow x = 0$

$x - 4 = 0 \Rightarrow x = 4$

Now, we can plot these values on a number line:

luaCopy code
    |------|------|------|
    -inf    0      4     +inf

Next, we choose test points in each interval and evaluate the inequality:

1. Test $x = -1$: $5(-1)(-1 - 4) \leq 0$ $\Rightarrow$ $25 \leq 0$ (False)
2. Test $x = 1$: $5(1)(1 - 4) \leq 0$ $\Rightarrow$ $-15 \leq 0$ (True)
3. Test $x = 5$: $5(5)(5 - 4) \leq 0$ $\Rightarrow$ $25 \leq 0$ (False)

The solution to the inequality $5x^2 - 20x \leq 0$ is $x \leq 0$ or $x \geq 4$.

2. $x^2 \geq 16$

To solve this inequality, we first factor it:

$x^2 - 16 \geq 0$

Now, we can factor the left-hand side:

$(x - 4)(x + 4) \geq 0$

To find the values of $x$ that satisfy the inequality, we set each factor to zero and solve for $x$:

$x - 4 = 0 \Rightarrow x = 4$

$x + 4 = 0 \Rightarrow x = -4$

Now, let's plot these values on a number line:

luaCopy code
    |------|------|------|
    -inf   -4      4     +inf

Next, we choose test points in each interval and evaluate the inequality:

1. Test $x = -5$: $(-5 - 4)(-5 + 4) \geq 0$ $\Rightarrow$ $-9 \geq 0$ (False)
2. Test $x = 0$: $(0 - 4)(0 + 4) \geq 0$ $\Rightarrow$ $-16 \geq 0$ (False)
3. Test $x = 5$: $(5 - 4)(5 + 4) \geq 0$ $\Rightarrow$ $9 \geq 0$ (True)

The solution to the inequality $x^2 \geq 16$ is $x \leq -4$ or $x \geq 4$.

3. $x^2 + 14x + 49 \leq 0$

To solve this inequality, we first check if it can be factored. The expression $x^2 + 14x + 49$ is a perfect square trinomial: $(x + 7)^2$. Therefore, the inequality can be rewritten as:

$(x + 7)^2 \leq 0$

Since the square of any real number is non-negative, the expression $(x + 7)^2$ is always greater than or equal to zero. The only way it can be less than or equal to zero is if the expression itself is zero:

$x + 7 = 0 \Rightarrow x = -7$

Now, we can plot this value on a number line:

luaCopy code
    |------|------|------|
    -inf   -7     +inf

Next, we choose a test point, $x = 0$, and evaluate the inequality:

$(0 + 7)^2 \leq 0$ $\Rightarrow$ $49 \leq 0$ (False)

Since the inequality is false for all values of $x$, there is no solution to $x^2 + 14x + 49 \leq 0$.

In summary:

1. The solution to $5x^2 - 20x \leq 0$ is $x \leq 0$ or $x \geq 4$.
2. The solution to $x^2 \geq 16$ is $x \leq -4$ 0 0