Certainly! Let's analyze the question step by step.
The problem is asking about an inequality of the form:
[tex]\[ ax^2 + bx + c > 0 \][/tex]
[tex]\[ ax^2 + bx + c < 0 \][/tex]
[tex]\[ ax^2 + bx + c \geq 0 \][/tex]
[tex]\[ ax^2 + bx + c \leq 0 \][/tex]
Here's what each of these forms means:
- [tex]\( ax^2 + bx + c > 0 \)[/tex] means the quadratic expression is greater than zero.
- [tex]\( ax^2 + bx + c < 0 \)[/tex] means the quadratic expression is less than zero.
- [tex]\( ax^2 + bx + c \geq 0 \)[/tex] means the quadratic expression is greater than or equal to zero.
- [tex]\( ax^2 + bx + c \leq 0 \)[/tex] means the quadratic expression is less than or equal to zero.
Breaking down the options:
- Linear Equation (a.) is an equation of the first degree, typically written in the form [tex]\( ax + b = 0 \)[/tex]. Examples include [tex]\( 2x + 3 = 0 \)[/tex]. This does not fit our inequality forms.
- Linear Inequality (b.) involves first-degree expressions and is written in forms like [tex]\( ax + b > 0 \)[/tex] or [tex]\( ax + b < 0 \)[/tex]. Again, this doesn't match our inequality forms.
- Quadratic Equation (c.) describes an equation of the second degree, written as [tex]\( ax^2 + bx + c = 0 \)[/tex]. This involves an equality, not an inequality.
- Quadratic Inequality (d.) involves second-degree expressions and inequalities, exactly matching our given forms.
From the detailed analysis, we see that the inequality described in the question fits the definition of a Quadratic Inequality.
Therefore, the correct answer is:
[tex]\[ \boxed{d. \text{Quadratic Inequality}} \][/tex]
