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In the first three parts, rearrange linear inequalities to make $x$ the subject.

In the last four parts, correctly give the direction of the inequality sign after rearranging an inequality.

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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}

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}

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Solve the following linear inequalities by finding the set of possible values for $x$. State your answers as fractions where applicable.

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$\\simplify{x-{a[0]}<{a[1]}}$

$x<$ [[0]]

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$\\simplify{{a[2]}x<{a[3]}}$

$x<$ [[0]]

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$\\simplify{{a[6]}x-{a[4]}<{a[5]}}$

$x<$ [[0]]

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$\\simplify{{-a[6]}x-{a[4]}<{a[5]}}$

$x$ [[1]] [[0]]

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$\\simplify{{b[0]}x-{b[1]}<{b[3]}-{b[2]}x}$

$x$ [[1]] [[0]]

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$\\simplify{-{b[4]}x+{a[8]}a>{b[5]}+b-{a[7]}x}$

$x$ [[1]] [[0]]

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$\\simplify{-{c}(x+g)>6h-{c}{a[0]}}$

$x$ [[1]] [[0]]

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