Answer
$$(-∞,\frac{-4-\sqrt{71}}{5}]∪[\frac{-4+\sqrt{71}}{5},+∞)$$
Work Step by Step
Let $5x^2 +8x \geq 11 $ be a function $f$.
Express the inequality in the form $f(x) \geq 0$. Begin by the rewriting the inequality so that $0$ is on the right side.
$5x^2 +8x \geq 11$
$5x^2 +8x -11\geq 11-11$
$5x^2 +8x -11\geq 0 $
Find the $x$-intercepts by solving $5x^2 +8x -11=0$
Use the quadratic formula: $x = \frac{-b±\sqrt{b^2-4ac}}{2a}$ where $a=5$, $b=8$, $c=-11$
$x = \frac{-8±\sqrt{8^2-4⋅5⋅-11}}{2⋅5}$
$x = \frac{-4-\sqrt{71}}{5}$ or $\frac{-4+\sqrt{71}}{5}$
These $x$-intercepts serve as boundary points that separate the number line into intervals.
The boundary points of this equation therefore divide the number line into three intervals:
$(-∞, \frac{-4-\sqrt{71}}{5})(\frac{-4-\sqrt{71}}{5},\frac{-4+\sqrt{71}}{5})(\frac{-4+\sqrt{71}}{5}, +∞)$
Choose one test value within each interval and evaluate $f$ at that number.
$(-∞, \frac{-4-\sqrt{71}}{5})$
Test Value = $-3$
$f=5x^2 +8x -11$
$f=5(-3)^2 +8(-3) -11$
$f=10$
Conclusion: $f (x) > 0$ for all $x$ in $(-∞,\frac{-4-\sqrt{71}}{5})$.
$(\frac{-4-\sqrt{71}}{5}, \frac{-4+\sqrt{71}}{5})$
Test Value = $0$
$f=5x^2 +8x -11$
$f=5(0)^2 +8(0) -11$
$f=-11$
Conclusion: $f (x) < 0$ for all $x$ in $(\frac{-4-\sqrt{71}}{5}, \frac{-4+\sqrt{71}}{5})$.
$(\frac{-4+\sqrt{71}}{5},+∞)$
Test Value = $1$
$f=5x^2 +8x -11$
$f=5(1)^2 +8(1) -11$
$f=2$
Conclusion: $f (x) > 0$ for all $x$ in $(\frac{-4+\sqrt{71}}{5},+∞)$.
Write the solution set, selecting the interval or intervals that satisfy the given inequality. We are interested in solving $f (x) \gt 0$, where $f (x) = 5x^2 +8x -11$. Based on the solution above, $f(x)\geq0$ for all $x$ in $(\frac{-4-\sqrt{71}}{5},-∞)$ or $(\frac{-4+\sqrt{71}}{5},+∞)$. However, because the inequality involves $\geq$ (greater than or equal to), we must also include the solutions of $5x^2+8x-11$, namely $\frac{-4-\sqrt{71}}{5}$ and $\frac{-4+\sqrt{71}}{5}$, in the solution set.
Thus, the solution set of the given inequality is:
$$(-∞,\frac{-4-\sqrt{71}}{5}]∪[\frac{-4+\sqrt{71}}{5},+∞)$$
