Question

Solve the inequality

$\frac{(x-2)^{10000}(x+1)^{253}(x-\frac{1}{2})^{971}(x+8)^4}{x^{500}(x-3)^{75}(x+2)^{93}} \ge 0$

Answer 1

$(-\infty, -2) \cup [-1, 0) \cup (0, \frac{1}{2}] \cup \{2\} \cup (3, \infty)$

Answer 2

To solve the inequality $\frac{(x-2)^{10000}(x+1)^{253}(x-\frac{1}{2})^{971}(x+8)^4}{x^{500}(x-3)^{75}(x+2)^{93}} \ge 0$, we use the sign chart method.

1. Find the Critical Points:

Set each factor in the numerator and denominator to zero:

• From numerator:
  • $x-2=0 \Rightarrow x=2$ (even power: 10000)
  • $x+1=0 \Rightarrow x=-1$ (odd power: 253)
  • $x-\frac{1}{2}=0 \Rightarrow x=\frac{1}{2}$ (odd power: 971)
  • $x+8=0 \Rightarrow x=-8$ (even power: 4)
• From denominator:
  • $x=0 \Rightarrow x=0$ (even power: 500)
  • $x-3=0 \Rightarrow x=3$ (odd power: 75)
  • $x+2=0 \Rightarrow x=-2$ (odd power: 93)

List all critical points in increasing order: $-8, -2, -1, 0, \frac{1}{2}, 2, 3$.

2. Analyze the Effect of Powers:

• Odd powers: At roots corresponding to factors with odd powers, the sign of the expression changes. These are $x=-1, x=\frac{1}{2}, x=-2, x=3$.
• Even powers: At roots corresponding to factors with even powers, the sign of the expression does not change. These are $x=2, x=-8, x=0$.
• Denominator roots: Points $x=0, x=3, x=-2$ make the denominator zero, so the expression is undefined at these points. They must always be excluded from the solution.
• Numerator roots: Points $x=2, x=-1, x=\frac{1}{2}, x=-8$ make the expression zero. Since the inequality is $\ge 0$, these points are included in the solution if they are not denominator roots.

3. Determine the Sign in Intervals (Sign Chart):

Start with the rightmost interval $(3, \infty)$. Let $x=4$:

All factors $(x-2), (x+1), (x-\frac{1}{2}), (x+8), x, (x-3), (x+2)$ are positive.

So, $\frac{(+)^{10000}(+)^{253}(+)^{971}(+)^4}{(+)^{500}(+)^{75}(+)^{93}} = (+) \ge 0$.

Thus, $f(x) > 0$ for $x \in (3, \infty)$.

Now, move left across the critical points, changing sign only at odd-powered roots:

Interval	Test Point	Sign of $f(x)$	Reason for sign change/no change
$(3, \infty)$	$x=4$	$+$	(Starting point)
$x=3$		Undefined	Odd power (75), denominator root
$(2, 3)$	$x=2.5$	$-$	Sign changes at $x=3$
$x=2$		$0$	Even power (10000), numerator root. Sign does not change across $x=2$.
$(\frac{1}{2}, 2)$	$x=1$	$-$	Sign does not change at $x=2$
$x=\frac{1}{2}$		$0$	Odd power (971), numerator root. Sign changes across $x=\frac{1}{2}$.
$(0, \frac{1}{2})$	$x=0.1$	$+$	Sign changes at $x=\frac{1}{2}$
$x=0$		Undefined	Even power (500), denominator root. Sign does not change across $x=0$.
$(-1, 0)$	$x=-0.5$	$+$	Sign does not change at $x=0$
$x=-1$		$0$	Odd power (253), numerator root. Sign changes across $x=-1$.
$(-2, -1)$	$x=-1.5$	$-$	Sign changes at $x=-1$
$x=-2$		Undefined	Odd power (93), denominator root. Sign changes across $x=-2$.
$(-8, -2)$	$x=-3$	$+$	Sign changes at $x=-2$
$x=-8$		$0$	Even power (4), numerator root. Sign does not change across $x=-8$.
$(-\infty, -8)$	$x=-9$	$+$	Sign does not change at $x=-8$

4. Formulate the Solution Set:

We need $f(x) \ge 0$. This means $f(x) > 0$ or $f(x) = 0$.

• Intervals where $f(x) > 0$: $(-\infty, -8)$, $(-8, -2)$, $(-1, 0)$, $(0, \frac{1}{2})$, $(3, \infty)$.

• Points where $f(x) = 0$ (numerator roots): $x=-8, x=-1, x=\frac{1}{2}, x=2$.

• Points where $f(x)$ is undefined (denominator roots): $x=-2, x=0, x=3$. These must be excluded.

Combine the intervals and points:

1. From $(-\infty, -8)$ and $x=-8$ and $(-8, -2)$: This combines to $(-\infty, -2)$. (Note: $x=-2$ is excluded).
2. From $(-1, 0)$ and $x=-1$: This combines to $[-1, 0)$. (Note: $x=0$ is excluded).
3. From $(0, \frac{1}{2})$ and $x=\frac{1}{2}$: This combines to $(0, \frac{1}{2}]$. (Note: $x=0$ is excluded).
4. The point $x=2$ makes $f(x)=0$, and it's not excluded by the denominator. So, $x=2$ is an isolated solution.
5. From $(3, \infty)$: This is an interval where $f(x) > 0$. (Note: $x=3$ is excluded).

The final solution is the union of these parts:

$x \in (-\infty, -2) \cup [-1, 0) \cup (0, \frac{1}{2}] \cup \{2\} \cup (3, \infty)$.