Animated Solution for Mathematics - Basic Mathematics: If $S$ is the set of all real $x$ such that $\frac{2x-1}{2x^3+3x^2+x}$ is positive, then $S$ contains

We are tasked with solving the rational inequality:

Our objective is to determine the set $S$ of all real values of $x$ that satisfy this condition. While the numerator is a simple linear term, the denominator is a cubic polynomial that requires careful factorization.

First, we extract the common factor $x$ from the denominator:

Next, we factor the quadratic expression $2x^2+3x+1$ by splitting the middle term. We look for two numbers that multiply to $2$ and add to $3$, which are $2$ and $1$:

Substituting these factors back into the original inequality, we obtain the fully factored form:

To apply the Wavy Curve method, we identify the critical points where the expression is either zero or undefined. Setting each factor to zero, we find:

1. $2x-1=0⟹x=\frac{1}{2}$ 2. $x=0$ 3. $2x+1=0⟹x=-\frac{1}{2}$ 4. $x+1=0⟹x=-1$

We arrange these critical points on the number line in ascending order: $-1,-\frac{1}{2},0,\frac{1}{2}$. These points divide the real number line into five distinct intervals.

We determine the sign of the expression in each interval. Starting from the rightmost interval ($x>\frac{1}{2}$), we test a value like $x=10$, which yields a positive result.

Since every factor in the expression has an odd exponent (power of $1$), the sign of the expression alternates as we cross each critical point. Moving from right to left, the signs are:

The inequality requires the expression to be strictly greater than zero. Selecting the intervals marked with a positive sign, we define the solution set $S$ as:

Conclusion: Any interval contained within these three regions represents a valid subset of the solution. For instance, $(-\infty ,-1.5)$ and $(\frac{1}{2},3)$ are valid subsets, as they lie entirely within the regions defined by $S$.