To solve the quadratic inequality [tex]\(6x^2 + 1 \leq 0\)[/tex], we can follow these logical steps:
1. Understand the quadratic expression:
- The left-hand side of the inequality is [tex]\(6x^2 + 1\)[/tex].
2. Analyze the expression:
- The term [tex]\(6x^2\)[/tex] is always non-negative since the square of any real number [tex]\(x\)[/tex] is non-negative and multiplying by 6 (a positive number) does not change that.
- Adding 1 to any non-negative number yields a result that is strictly positive.
3. Check the inequality:
- For [tex]\(6x^2 + 1\)[/tex] to be less than or equal to 0, the expression [tex]\(6x^2 + 1\)[/tex] would have to be negative or zero.
- Since [tex]\(6x^2 + 1\)[/tex] is strictly positive for any real value of [tex]\(x\)[/tex], this condition [tex]\(6x^2 + 1 \leq 0\)[/tex] can never be satisfied.
Given these observations, the inequality [tex]\(6x^2 + 1 \leq 0\)[/tex] has no real solutions. Therefore, the solution set is the empty set [tex]\(\varnothing\)[/tex].
So the correct answer is: [tex]\(\varnothing\)[/tex].
