Graphing $\left(\;(4x -2 \leq 6 \;\text{ and }\; 3-x \leq 7)\;\text{ or }\;(3x > 18 \;\text{ and }\; 2x-9 \geq 11)\;\right)$ [closed]

Question

This is a homework question from a precalculus class that I'm a TA for.

Graph on a real line all points $x$ such that \begin{equation}\tag{$\star$} (4x -2 \leq 6 \;\text{ and }\; 3-x \leq 7) \;\;\text{ or }\;\; (3x > 18 \;\text{ and }\; 2x-9 \geq 11)\,. \end{equation}

Enough students were struggling with it, especially with how to mathematically interpret "and" and "or", that I wanted to write up a thorough solution to this exercise for my class, and figured I'd post it online to help anyone else who may wander across it.

Answer 1

The four inequalities above can be written more simply as the following four inequalities: \begin{equation}\tag{$\star$} (x \leq 2 \;\text{ and }\; x \geq -4) \;\;\text{ or }\;\; (x > 6 \;\text{ and }\; x \geq 10) \end{equation} The word "and" between two inequalities indicates that we need them to simultaneously be true, which corresponds to taking the intersection of their individual solution sets. So the first pair of inequalities are true for all $x$ such that $-4 \leq x \leq 2$. On a number line, this looks like:

Similarly for the other pair of inequalities, the intersection of their solution sets are all $x$ such that $x \geq 10$.

Now the word "or" between two inequalities indicates that we just need either one of them to be true, which corresponds to taking the union of their solution sets. So all the points $x$ that satisfy $(\star)$, when graphed on a number line, would look like