You should now understand the difference between equalities $x = 5$ and inequalities $ x < 2$, where the first expression only has one answer \fillin[5], and the other has an infinite amount of answers: \fillin[any number less than 2].

We can now add another layer of complexity to this and learn how to do some mathematical operations on on inequality expression.

Let us begin with $-1 < 5$, which is something we agree to be true.

From here we can add 3 to both sides yielding \fillin[$2 < 8$]. We agree that this is also true!

What about subtraction? Let's continue with our inequality from the last blank and subtract 1: $1 < 7$. That still checks out, very cool.

How about we multiply the whole expression by $-1$?
\[-1 < -7\]
Is this still true?...

\fillin[NO]!

That is the most important thing to remember when manipulating inequalities!
\begin{theorembox}
    \textbf{When multiplying an inequality by a negative number you must flip the inequality!}
\end{theorembox}


\begin{examplebox}
    Let us practise our 4 operations of addition, subtraction, multiplication and division on inequality expressions:
    \begin{questions}
        \Question[2]
        \begin{parts}
            \part Solve the inequality \(4x-5 < 3\)
            \part Graph the solution set on the number line\\
            \begin{center}
            \begin{tikzpicture}
                \draw[latex-latex] (-6,0) -- (4,0);
                \foreach \x in {-5,...,3}
                    \draw (\x,0.1) -- (\x,-0.1) node[below] {\x};

                \ifprintanswers
                \draw[very thick, blue, -{Stealth[scale=1.5]}] (1.9,0.5) -- (-4.1,0.5);
                \filldraw[fill=white, draw=blue, very thick] (2,0.5) circle (2pt);
                \fi
            \end{tikzpicture}
            \end{center}
        \end{parts}
        \Question[2] Solve each of the following inequalities:
        \begin{parts}\begin{multicols}{2}
            \part \[-2x \leq 6\]
            \begin{solutionordottedlines}[1in]
                \(-2 x \leq 6\)
                \(x \geq-3\) (Divide both sides by -2 and reverse the inequality sign.)
            \end{solutionordottedlines}
            \part \[-\frac{x}{3}>4\]
            \begin{solutionordottedlines}[1in]
                \(-\frac{x}{3}>4\)
                \(x<-12\) (Multiply both sides by -3 and reverse the inequality sign.)
            \end{solutionordottedlines}
        \end{multicols}\end{parts}
        \Question[2]
        \begin{parts}
            \part Solve the inequality \(2 x+3<3 x-4\).
            \begin{solutionordottedlines}[1in]
                \subsubsection*{Method 1}
                    \[
                    \begin{aligned}
                    2 x+3 & <3 x-4 & & \\
                    3 & <x-4 & & \text { (Subtract } 2 x \text { from both sides.) } \\
                    7 & <x & & \text { (Add } 4 \text { to both sides.) } \\
                    \text { so } x & >7 & &
                    \end{aligned}
                    \]
                \subsubsection*{Method 2}
                    \[
                    \begin{aligned}
                    2 x+3 & <3 x-4 \\
                    2 x & <3 x-7 \\
                    -x & <-7 \\
                    x & >7
                    \end{aligned}
                    \]

                    (Subtract 3 from both sides.)

                    (Subtract \(3 x\) from both sides.)

                    (Multiply by -1 and reverse the inequality sign.)

                    The solution set is \(\{x: x>7\}\).
            \end{solutionordottedlines}
            \part Graph the solution set on the number line.\\
            \begin{center}
            \begin{tikzpicture}
                \draw[latex-latex] (-1,0) -- (9,0);
                \foreach \x in {0,...,8}
                    \draw (\x,0.1) -- (\x,-0.1) node[below] {\x};

                \ifprintanswers
                \draw[very thick, blue, -{Stealth[scale=1.5]}] (7.1,0.5) -- (8.5,0.5);
                \filldraw[fill=white, draw=blue, very thick] (7,0.5) circle (2pt);
                \fi
            \end{tikzpicture}
            \end{center}
        \end{parts}
    \end{questions}
\end{examplebox}

\begin{exercisebox}
    \subsection{Exercises:}
    \begin{questions}
        \Question[6] Solve each of these inequalities. Graph each solution on a number line.
        \begin{parts}
            \part \[x+3 \geq 7\]
            \begin{solutionorbox}[1in]
            \end{solutionorbox}
            \part \[x-10 > -6\]
            \begin{solutionorbox}[1in]
            \end{solutionorbox}
            \part \[3x>-15\]
            \begin{solutionorbox}[1in]
            \end{solutionorbox}
        \end{parts}
        \Question[3] Solve the following purely algebraically:
        \begin{parts}
            \part \[2x+1 \geq 5\]
            \begin{solutionordottedlines}[1in]
            \end{solutionordottedlines}
            \part \[\frac{4x}{7}\leq -2\]
            \begin{solutionordottedlines}[1in]
            \end{solutionordottedlines}
            \part \[\frac{x}{3}-\frac{1}{2} \geq 1\]
            \begin{solutionordottedlines}[1in]
            \end{solutionordottedlines}
        \end{parts}
        \Question[3] Solve: (don't forget to reverse the inequality sign when dividing the expression through by a negative number!)
        \begin{parts}
            \part \[-4 x \leq 20\]
            \begin{solutionordottedlines}[1in]
            \end{solutionordottedlines}
            \part \[-\frac{x}{2} \leq 5\]
            \begin{solutionordottedlines}[1in]
            \end{solutionordottedlines}
            \part \[-\frac{x}{12} \geq-8\]
            \begin{solutionordottedlines}[1in]
            \end{solutionordottedlines}
        \end{parts}
        \Question[2] Solve:
        \begin{parts}
            \part \[3-2 x>5\]
            \begin{solutionordottedlines}[1in]
            \end{solutionordottedlines}
            \part \[6-\frac{2 x}{3}<4\]
            \begin{solutionordottedlines}[1in]
            \end{solutionordottedlines}
            \part \[\frac{x+3}{2} \leq \frac{3-x}{2}\]
            \begin{solutionordottedlines}[1in]
            \end{solutionordottedlines}
        \end{parts}
        \Question[5] Solve:
        \begin{parts}
            \part \[1.2 x+6.8 \leq 15.2\]
            \begin{solutionordottedlines}[1in]
            \end{solutionordottedlines}
            \part \[1.6(x+7) \leq 1.5(x-3)\]
            \begin{solutionordottedlines}[1in]
            \end{solutionordottedlines}
            \part \[2 x-14 \leq 8\]
            \begin{solutionordottedlines}[1in]
            \end{solutionordottedlines}
            \part \[\frac{2 x+1}{6}>-3\]
            \begin{solutionordottedlines}[1in]
            \end{solutionordottedlines}
            \part \[\frac{x-8}{2}-\frac{2 x}{3} \geq 3\]
            \begin{solutionordottedlines}[1in]
            \end{solutionordottedlines}
        \end{parts}
        \Question[2] When 5 is added to twice \(p\), the result is greater than 17 . What values can \(p\) take?
            \begin{solutionordottedlines}[1in]
            \end{solutionordottedlines}
        \Question[2] When 16 is subtracted from half of \(q\), the result is less than 18 . What values can \(q\) take?
            \begin{solutionordottedlines}[1in]
            \end{solutionordottedlines}
        \Question[2] When \(2 p\) is subtracted from 10 , the result is greater than or equal to 4 . What values can \(p\) take?
            \begin{solutionordottedlines}[1in]
            \end{solutionordottedlines}
    \end{questions}
\end{exercisebox}
