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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.

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For example, the following inequality is true:

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\$-3 \\lt -2 \$

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

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\$6 \\gt 4 \$

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#### a)

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

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

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#### b)

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

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

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#### c)

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

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#### d)

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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.

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

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#### e)

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

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

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#### f)

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

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

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g)

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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.

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

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In parts (a) and (b) rearrange linear inequalities to make $x$ the subject.

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In the parts (c) and (d) correctly give the direction of the inequality sign after rearranging an inequality.

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$\\simplify{{a}x<{a}}$

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$x<$ []

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>

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<

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$\\simplify{{b}x-{b}<{b}-{b}x}$

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$x$  []  []

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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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