As with regular linear equations, we aim to isolate the variable by subtracting any constants when dividing by the $x$ coefficient. The only major difference is that when we divide or multiply by a negative number, the inequality sign is reversed.

For example, the following inequality is true:

\[ -3 \lt -2 \]

When we multiply both sides by $-2$, the inequality sign must reverse:

\[ 6 \gt 4 \]

a)

To put $x$ on its own, we need to add $\var{a[0]}$ to both sides of the inequality.

\begin{align}
\simplify{x-{a[0]}}&<\var{a[1]}\\[1em]
\var{x}&<\simplify[]{{a[1]}+{a[0]}}\\[1em]
x&<\simplify{({a[1]}+{a[0]})}\text{.}
\end{align}

b)

In this example we find $x$ by dividing both sides by the coefficient of $x$, $\var{a[2]}$.

\begin{align}
\simplify{{a[2]}}x&<\var{a[3]}\\[1em]
x&<\simplify{{a[3]}/{a[2]}}\text{.}
\end{align}

c)

\begin{align}
\simplify{{a[6]}x-{a[4]}}&<\var{a[5]}\\[1em]
\var{a[6]}x&<\var{a[5]}+\var{a[4]} & \text{Add } 8 \text{ to get } x \text{ on its own.}\\[1em]
x&<\simplify[]{({a[5]}+{a[4]})/{a[6]}} & \text{ Divide by } \var{a[6]} \text{.} \\[1em]
x&<\simplify{({a[5]}+{a[4]})/{a[6]}}\text{.}
\end{align}

d)

In this example, take the constants to one side, and keep the $x$ term on the other. Divide through by the negative $x$-coefficient to find an inequality for $x$. Notice that where you divide (or multiply) an equality by a negative value, the inequality sign is reversed.

\begin{align}
\simplify{{-a[6]}x - {a[4]}} &< \var{a[5]} \\[1em]
\var{-a[6]}x &< \var{a[5]} + \var{a[4]} & \text{Add } \var{a[4]} \text{ to both sides.} \\[1em]
x &> \simplify[]{({a[5]}+{a[4]})/-{a[6]}} \text{ Divide by } \var{-a[6]} \text{. The inequality is reversed.} \\[1em]
x &> \simplify{({a[5]}+{a[4]})/-{a[6]}}\text{.}\\
\end{align}

e)

In this example, separate the constants and the $x$-term, then divide by the $x$-coefficient to find an inequality for $x$.

\begin{align}
\simplify{{b[0]}x-{b[1]}}&<\simplify{{b[3]}-{b[2]}x}\\[1em]
\simplify{({b[0]}+{b[2]})x}&<\simplify{{b[3]}+{b[1]}}\\[1em]
x&<\simplify{({b[3]}+{b[1]})/({b[0]}+{b[2]})}\text{.}\\[1em]
\end{align}

f)

In this example, separate the $x$-term from all other terms and remember to reverse the inequality when dividing by $\simplify{{a[7]}-{b[4]}}$.

\begin{align}
\simplify{-{b[4]}x+{a[8]}a}&>\simplify{{b[5]}+b-{a[7]}x}\\[1em]
\simplify{{a[7]}-{b[4]}}x&>\simplify{{b[5]}+b-{a[8]}a}\\[1em]
x&<\simplify{(-{b[5]}-b+{a[8]}a)/({b[4]}-{a[7]})}\text{.}\\[1em]
\end{align}

g)

In this example, a simple way to solve for $x$ is to divide by $-\var{c}$ before rearranging the rest of the equation by subtracting $g$ from both sides.

\begin{align}
\simplify{-{c}(x+g)}&>\simplify{6h-{c}{a[0]}}\\[1em]
\simplify{(x+g)}&<\simplify[]{6h/-{c}+{a[0]}}\\[1em]
x&<\simplify[]{6h/-{c}+{a[0]}-g}\\[1em]
x&<\simplify{{a[0]}-6h/{c}-g}\text{.}
\end{align}