Answer
$[-1, 1]\cup\left\{\frac{7}{2}\right\}$
Work Step by Step
1. Express the inequality in the form $f(x)<0, f(x)>0, f(x)\leq 0$, or $f(x)\geq 0,$
where $f$ is a polynomial function.
$(2x-7)^4(x-1)^3(x+1) \leq 0$
$f(x)=(2x-7)^4(x-1)^3(x+1)$
2. Solve the equation $f(x)=0$. The real solutions are the boundary points.
$(2x-7)^4(x-1)^3(x+1)=0$
$x=\frac{7}{2}$ or $x=1$ or $x=-1$
3. Make a table or diagram: use the test values to make a table or diagram of the sign of each factor in each interval.
4. Test each interval's sign of $f(x)$ with a test value,
$\begin{array}{llll}
Intervals: & a=test.v. & f(a),signs & f(a) \leq 0 ? \\
& &(2a-7)^4(a-1)^3(a+1) & \\
(-\infty,-1) & -4 & (+)(-)(-) & F\\
(-1,1) & 0 & (+)(-)(+) & T\\
(1,\frac{7}{2}) & 2 & (+)(+)(+) & F\\
(\frac{7}{2},\infty) & 4 & (+)(+)(+) & F
\end{array}$
5. Write the solution set, selecting the interval or intervals that satisfy the given inequality. If the inequality involves $\leq$ or $\geq$, include the boundary points.
Solution set: $[-1, 1]\cup\left\{\frac{7}{2}\right\}$
