Answer
$$[-3, 7]$$
Work Step by Step
Let $(x - 7)(x + 3)\leq0$ be a function $f$.
Find the $x$-intercepts by solving $(x - 7)(x + 3)=0$
$x-7=0$ or $x+3=0$
$x = 7$ or $x = -3$
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:
$(-∞, -3)(-3,7)(7, +∞)$
Choose one test value within each interval and evaluate $f$ at that number.
$(-∞, -3)$
Test Value = $-4$
$f=(x - 7)(x + 3)$
$f=(-4 - 7)(-4 + 3)$
$f=11$
Conclusion: $f (x) > 0$ for all $x$ in $(-∞, -3)$.
$(-3, 7)$
Test Value = $0$
$f=(x - 7)(x + 3)$
$f=(0 - 7)(0 + 3)$
$f=-21$
Conclusion: $f (x) < 0$ for all $x$ in $(-3,7)$.
$(7,+∞)$
Test Value = $8$
$f=(x - 7)(x + 3)$
$f=(8 - 7)(8 + 3)$
$f=11$
Conclusion: $f (x) > 0$ for all $x$ in $(7,+∞)$.
Write the solution set, selecting the interval or intervals that satisfy the given
inequality. We are interested in solving $f (x) \leq 0$, where $f (x) = (x - 7)(x + 3)$. Based on the solution above, $f(x)\leq0$ for all $x$ in $(-3, 7)$. However, because the inequality involves $\leq$ (less than or equal to), we must also include the solutions of $(x - 7)(x + 3) = 0$, namely $-3$ and $7$, in the solution set.
Thus, the solution set of the given inequality is:
$$[-3, 7]$$
