College Algebra, Chapter 1, 1.6, Section 1.6, Problem 44

Solve the nonlinear inequality $5x^2 + 3x \geq 3x^2 + 2$. Express the solution using interval notation and graph the solution set.

$
\begin{equation}
\begin{aligned}
5x^2 + 3x & \geq 3x^2 + 2\\
\\
5x^2 - 3x^2 + 3x - 2 & \geq 0 && \text{Subtract } 3x^2 \text{ and } 2\\
\\
2x^2 + 3x - 2 & \geq 0 && \text{Factor}\\
\\
(x+2)(2x-1) & \geq 0
\end{aligned}
\end{equation}
$


The factors on the left side are $x+2$ and $2x - 1$. These factors are zero when $x$ is -2 and $\displaystyle \frac{1}{2}$ respectively. These numbers divide the real line into intervals.
$\displaystyle (-\infty, -2], \left( -2, \frac{1}{2} \right), \left[ \frac{1}{2},\infty \right)$



From the diagram, the volutions of the inequality $5x^2 + 3x \geq 3x^2 + 2$
are $(-\infty, -2] \bigcup \left[\frac{1}{2}, \infty \right)$