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Given $\\var{a}x+\\var{b}<\\var{c}$, solving for $x$ gives $x<$ [[0]].
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\n$\\var{a}x+\\var{b}$ | \n$<$ | \n$\\var{c}$ | \n
\n | \n | \n |
$\\var{a}x+\\var{b}-\\var{b}$ | \n$<$ | \n$\\var{c}-\\var{b}$ | \n
\n | \n | \n |
$\\var{a}x$ | \n$<$ | \n$\\var{c-b}$ | \n
\n | \n | \n |
$\\displaystyle{\\frac{\\var{a}x}{\\var{a}}}$ | \n$<$ | \n$\\displaystyle{\\frac{\\var{c-b}}{\\var{a}}}$ | \n
\n | \n | \n |
$x$ | \n$<$ | \n$\\displaystyle{\\simplify{{c-b}/{a}}}$ | \n
To satisfy $\\var{d}-\\var{f}y\\le\\var{g}$, we require $y\\ge$ [[0]].
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\n\n$\\var{d}-\\var{f}y$ | \n$\\le$ | \n$\\var{g}$ | \n
\n | \n | \n |
$\\var{d}-\\var{f}y-\\var{d}$ | \n$\\le$ | \n$\\var{g}-\\var{d}$ | \n
\n | \n | \n |
$-\\var{f}y$ | \n$\\le$ | \n$\\var{g-d}$ | \n
\n | \n | \n |
$\\displaystyle{\\frac{\\var{-f}y}{\\var{-f}}}$ | \n$\\ge$ | \n$\\displaystyle{\\frac{\\var{g-d}}{\\var{-f}}}$ | \n
\n | \n | \n |
$y$ | \n$\\ge$ | \n$\\displaystyle{\\simplify{{g-d}/{-f}}}$ | \n
Rearrange $\\displaystyle{-\\frac{z}{\\var{h}}}-\\var{j}<\\var{k}$ to determine the value of $z$.
\n$z>$ [[0]]
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\n$\\displaystyle{-\\frac{z}{\\var{h}}}-\\var{j}$ | \n$<$ | \n$\\var{k}$ | \n
\n | \n | \n |
$\\displaystyle{-\\frac{z}{\\var{h}}}-\\var{j}+\\var{j}$ | \n$<$ | \n$\\var{k}+\\var{j}$ | \n
\n | \n | \n |
$\\displaystyle{-\\frac{z}{\\var{h}}}$ | \n$<$ | \n$\\var{k+j}$ | \n
\n | \n | \n |
$\\displaystyle{-\\frac{z}{\\var{h}}\\times\\var{h}}$ | \n$<$ | \n$\\var{k+j}\\times \\var{h}$ | \n
\n | \n | \n |
$-z$ | \n$<$ | \n$\\var{-ans3}$ | \n
\n | \n | \n |
$z$ | \n$>$ | \n$\\var{ans3}$ | \n
Solve $\\displaystyle{\\frac{a-\\var{l}}{\\var{m}}}\\le\\var{n}$ for $a$.
\n$a\\le$ [[0]]
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\n$\\displaystyle{\\frac{a-\\var{l}}{\\var{m}}}$ | \n$\\le$ | \n$\\var{n}$ | \n
\n | \n | \n |
$\\displaystyle{\\frac{a-\\var{l}}{\\var{m}}}\\times \\var{m}$ | \n$\\le$ | \n$\\var{n}\\times\\var{m}$ | \n
\n | \n | \n |
$a-\\var{l}$ | \n$\\le$ | \n$\\var{n*m}$ | \n
\n | \n | \n |
$a-\\var{l}+\\var{l}$ | \n$\\le$ | \n$\\var{n*m}+\\var{l}$ | \n
\n | \n | \n |
$a$ | \n$\\le$ | \n$\\var{ans4}$ | \n
Solve $\\var{p}>\\var{q}(\\var{r}+b)$.
\n$b>$ [[0]]
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\n$\\var{p}$ | \n$>$ | \n$\\var{q}(\\var{r}+b)$ | \n
\n | \n | \n |
$\\displaystyle{\\frac{\\var{p}}{\\var{q}}}$ | \n$<$ | \n$\\displaystyle{\\frac{\\var{q}(\\var{r}+b)}{\\var{q}}}$ | \n
\n | \n | \n |
$\\displaystyle{\\simplify{{p}/{q}}}$ | \n$<$ | \n$\\var{r}+b$ | \n
\n | \n | \n |
$\\displaystyle{\\simplify{{p}/{q}}}-\\var{r}$ | \n$<$ | \n$\\var{r}+b-\\var{r}$ | \n
\n | \n | \n |
$\\displaystyle{\\simplify{{p-r*q}/{q}}}$ | \n$<$ | \n$b$ | \n
\n | \n | \n |
$b$ | \n$>$ | \n$\\displaystyle{\\simplify{{p-r*q}/{q}}}$ | \n
Solve $\\displaystyle{\\frac{\\var{s}w}{\\var{t}}}\\ge\\var{u}$.
\n$w\\le$ [[0]]
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\n$\\displaystyle{\\frac{\\var{s}w}{\\var{t}}}$ | \n$\\ge$ | \n$\\var{u}$ | \n
\n | \n | \n |
$\\displaystyle{\\frac{\\var{s}w}{\\var{t}}}\\times\\var{t}$ | \n$\\ge$ | \n$\\var{u}\\times\\var{t}$ | \n
\n | \n | \n |
$\\var{s}w$ | \n$\\ge$ | \n$\\var{u*t}$ | \n
\n | \n | \n |
$\\displaystyle{\\frac{\\var{s}w}{\\var{s}}}$ | \n$\\le$ | \n$\\displaystyle{\\frac{\\var{u*t}}{\\var{s}}}$ | \n
\n | \n | \n |
$w$ | \n$\\le$ | \n$\\displaystyle{\\simplify{{u*t}/{s}}}$ | \n