Answer
$$[-4, \frac{2}{3}]$$
Work Step by Step
Let $3x^2 + 10x - 8 \leq 0$ be a function $f$.
Find the $x$-intercepts by solving $3x^2 + 10x - 8=0$
Factor: $3x^2 + 10x - 8 = (3x-2)(x+4)$
$3x-2=0$ or $x+4=0$
$x = \frac{2}{3}$ or $x = -4$
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:
$(-∞, -4)(-4,\frac{2}{3})(\frac{2}{3}, +∞)$
Choose one test value within each interval and evaluate $f$ at that number.
$(-∞, -4)$
Test Value = $-4$
$f=(3x-2)(x+4)$
$f=(3(4)-2)(4+4)$
$f=80$
Conclusion: $f (x) > 0$ for all $x$ in $(-∞, -4)$.
$(-4, \frac{2}{3})$
Test Value = $0$
$f=(3x-2)(x+4)$
$f=(3(0)-2)(0+4)$
$f=-8$
Conclusion: $f (x) < 0$ for all $x$ in $(-4,\frac{2}{3})$.
$(\frac{2}{3},+∞)$
Test Value = $1$
$f=(3x-2)(x+4)$
$f=(3(1)-2)(1+4)$
$f=5$
Conclusion: $f (x) > 0$ for all $x$ in $(\frac{2}{3},+∞)$.
Write the solution set, selecting the interval or intervals that satisfy the given inequality. We are interested in solving $f (x) \leq 0$, where $f (x) = 3x^2+10x-8$. Based on the solution above, $f(x)\lt0$ for all $x$ in $(-4, \frac{2}{3})$. However, because the inequality involves $\leq$ (less than or equal to), we must also include the solutions of $3x^2+10x-8 = 0$, namely $\frac{2}{3}$ and $-4$, in the solution set.
Thus, the solution set of the given inequality is:
$$[-4, \frac{2}{3}]$$
