Sure, let's solve the inequality [tex]\( x^2 \sqrt{3} + 6x \leq 0 \)[/tex].
### Step 1: Factor the Inequality
First, notice that the inequality [tex]\( x^2 \sqrt{3} + 6x \leq 0 \)[/tex] can be factored to make it easier to solve.
We'll find the common factor [tex]\(x\)[/tex]:
[tex]\[ x (\sqrt{3} x + 6) \leq 0 \][/tex]
### Step 2: Analyze the Factors
Now, we need to consider the two factors of this inequality:
1. [tex]\( x \)[/tex]
2. [tex]\( \sqrt{3} x + 6 \)[/tex]
For the product of the two factors to be less than or equal to zero, one of them must be non-positive (less than or equal to zero) and the other must be non-negative (greater than or equal to zero).
### Step 3: Solve Each Factor
#### Factor 1: [tex]\( x \leq 0 \)[/tex]
[tex]\( x \)[/tex] must be less than or equal to zero.
#### Factor 2: [tex]\( \sqrt{3} x + 6 \geq 0 \)[/tex]
Isolate [tex]\(x\)[/tex]:
[tex]\[ \sqrt{3} x + 6 \geq 0 \][/tex]
[tex]\[ \sqrt{3} x \geq -6 \][/tex]
Divide by [tex]\(\sqrt{3}\)[/tex] (note that [tex]\(\sqrt{3}\)[/tex] is positive so the inequality direction remains the same):
[tex]\[ x \geq -\frac{6}{\sqrt{3}} \][/tex]
Rationalize the denominator:
[tex]\[ x \geq -2 \sqrt{3} \][/tex]
### Step 4: Combine the Solutions
Now, we need the intersection of the intervals [tex]\(x \leq 0\)[/tex] and [tex]\(x \geq -2\sqrt{3}\)[/tex].
These two conditions together mean:
[tex]\[ -2\sqrt{3} \leq x \leq 0 \][/tex]
### Conclusion
So, the solution to the inequality [tex]\( x^2 \sqrt{3} + 6 x \leq 0 \)[/tex] is:
[tex]\[ -2\sqrt{3} \leq x \leq 0 \][/tex]
Or in interval notation:
[tex]\[ x \in [-2\sqrt{3}, 0] \][/tex]
This is the range of values for [tex]\(x\)[/tex] that satisfy the given inequality.
