Solving inequalities advanced

"Advanced" inequalities are ones where you handle a range at once or combine several bounds, rather than solving a single simple inequality.
The rules are familiar, but you apply them to more than two pieces at a time.

A compound inequality traps a variable between two values, like $-2 < x + 2 < 3$.
Whatever you do, do it to all three parts: subtracting $2$ throughout gives $-4 < x < 1$.
The solution is the range of values that satisfy both ends at once.

You can combine two bounds to bound an expression built from them.
If $a > 2$ and $b > 3$, then multiplying the two positive bounds gives $ab > 6$, so the smallest $ab$ can approach is $6$.
The same idea gives a minimum for a sum: $a > 2$ and $b > 3$ make $a + b > 5$.

Word problems often hide a scaled bound.
If one item costs $s > 10$ dollars, then $5$ identical items cost $5s$, and multiplying the bound by $5$ gives $5s > 50$.
So the total must be more than $\$50$. Keep every sign's direction the same, and flip it only if you multiply or divide by a negative.

Solve the compound inequality $-1 \le 2x + 3 < 9$.
Subtract $3$ from all three parts: $-4 \le 2x < 6$.
Then divide every part by $2$: $-2 \le x < 3$.

If $a \ge 4$ and $b \ge 5$, what is the minimum possible value of $ab$?
Both bounds are positive, so $ab$ is smallest when $a$ and $b$ are at their smallest: $a = 4$ and $b = 5$.
So the minimum value of $ab$ is $4 \times 5 = 20$.

Each pencil costs more than $\$3$, and a student buys $4$ pencils. What can we say about the total cost?
If the price is $s > 3$, then $4$ pencils cost $4s$, and multiplying the bound by $4$ gives $4s > 12$.
So the total is more than $\$12$.