Answer
$$[-1, 7]$$
Work Step by Step
Let $(x + 1)(x - 7)\leq0$ be a function $f$.
Find the $x$-intercepts by solving $(x + 1)(x - 7)=0$
$x+1=0$ or $x-7=0$
$x = -1$ or $x = 7$
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:
$(-∞, -1)(-1,7)(7, +∞)$
Choose one test value within each interval and evaluate $f$ at that number.
$(-∞, -1)$
Test Value = $-2$
$f=(x + 1)(x - 7)$
$f=(-2 + 1)(-2 - 7)$
$f=9$
Conclusion: $f (x) > 0$ for all $x$ in $(-∞, -1)$.
$(-1, 7)$
Test Value = $0$
$f=(x + 1)(x - 7)$
$f=(0 + 1)(0 - 7)$
$f=-7$
Conclusion: $f (x) < 0$ for all $x$ in $(-1,7)$.
$(7,+∞)$
Test Value = $8$
$f=(x + 1)(x - 7)$
$f=(8 + 1)(8 - 7)$
$f=9$
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 $f (x) = (x + 1)(x - 7)$. Based on the solution above, $f(x)\leq0$ for all $x$ in $(-1, 7)$. However, because the inequality involves $\leq$ (less than or equal to), we must also include the solutions of $(x + 1)(x - 7) = 0$, namely $-1$ and $7$, in the solution set.
Thus, the solution set of the given inequality is:
$$[-1, 7]$$
