Solve the inequality. (Enter your answer using interval notation.) $$ \frac{4}{x-7}\frac{2}{x+4} $$ $$ \square $$

Question

Solve the inequality. (Enter your answer using interval notation.) $$ \frac{4}{x-7}>\frac{2}{x+4} $$ $$ \square $$

UpStudy AI · Accepted Answer

To solve the inequality

$$
\frac{4}{x-7} > \frac{2}{x+4},
$$

we first eliminate the fractions by cross-multiplying, which is valid as long as we keep track of the signs of the expressions involved. We rewrite the inequality as:

$$
4(x + 4) > 2(x - 7).
$$

Expanding both sides gives:

$$
4x + 16 > 2x - 14.
$$

Next, we isolate $$x$$ by subtracting $$2x$$ from both sides:

$$
4x - 2x + 16 > -14,
$$

which simplifies to:

$$
2x + 16 > -14.
$$

Now, subtract 16 from both sides:

$$
2x > -30.
$$

Dividing both sides by 2 yields:

$$
x > -15.
$$

Next, we need to consider the restrictions imposed by the original inequality. The expressions $$\frac{4}{x-7}$$ and $$\frac{2}{x+4}$$ are undefined when $$x = 7$$ and $$x = -4$$, respectively. Therefore, we must exclude these points from our solution.

Now we combine our findings. The solution $$x > -15$$ must be considered along with the restrictions:

1. $$x \neq 7$$
2. $$x \neq -4$$

Thus, we can express the solution in interval notation. The solution is:

$$
(-15, -4) \cup (-4, 7) \cup (7, \infty).
$$

So the final answer is:

$$
\boxed{(-15, -4) \cup (-4, 7) \cup (7, \infty)}.
$$
$$ (-15, -4) \cup (-4, 7) \cup (7, \infty). $$

Solve the inequality. (Enter your answer using interval notation.)

Solución de inteligencia artificial de Upstudy

Responder

Solución

Bonus Knowledge

preguntas relacionadas

Latest Algebra Questions