Animated Solution for Mathematics - Basic Mathematics: Find the set of all $x$ for which $\frac{2x}{2x^2+5x+2}>\frac{1}{x+1}$

The given inequality is:

The first instinct for many students is to cross-multiply. Stop! In the world of inequalities, cross-multiplication is a dangerous trap because the sign of the denominator $(x+1)$ is unknown, meaning we cannot determine if the inequality sign should flip or remain the same.

Instead, we must bring all terms to one side to maintain mathematical rigor.

The Anatomy of the Expression

Before moving terms, we factor the quadratic $2x^2+5x+2$. By splitting the middle term $5x$ into $4x+x$, we group the terms as:

Substituting this back into the inequality, we have:

Subtracting $\frac{1}{x+1}$ from both sides yields:

The Algebraic Dance

We now find the common denominator, which is $(2x+1)(x+2)(x+1)$. Combining the fractions, the numerator becomes:

Expanding these terms, we get $2x^2+2x$ for the first part and $2x^2+5x+2$ for the second. Subtracting them results in:

Our inequality is now:

Wavy Curve Mastery

To simplify the wavy curve method, we multiply the entire inequality by $-1$. Remember the golden rule: multiplying an inequality by a negative number flips the inequality sign:

The critical points where the numerator or denominator equals zero are $x=-2/3$, $x=-1/2$, $x=-2$, and $x=-1$. Sorting these in ascending order, we have:

Plotting these on a number line, we observe that since all factors have an odd power, the sign alternates across each interval. Starting from the rightmost interval ($x>-1/2$), the expression is positive.

Moving left, the signs alternate: positive, negative, positive, negative, positive. We seek the regions where the expression is less than zero.

The solution intervals are $(-2,-1)$ and $(-2/3,-1/2)$. Thus, the final solution is: