// Numbas version: exam_results_page_options {"name": "Inequalities involving a rational function/hyperbola", "extensions": ["jsxgraph"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"preamble": {"css": "", "js": ""}, "variables": {"c": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(-12..12 except [0,a])", "name": "c"}, "prod": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "leadCoeff*choice", "name": "prod"}, "strict": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "if(choice=1 or choice=3,true,false)", "name": "strict"}, "choice": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..4)", "name": "choice"}, "a": {"description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(-12..12 except 0)", "name": "a"}, "r2": {"description": "

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You could do this question by considering the cases (of the denominator being negative or positive) separately. However, the following method uses a nice trick to ensure you know the sign of what you are multiplying by.

Before we start manipulating the inequality, note that we can't divide by zero and hence $x\\ne \\var[basic,fractionNumbers,simplifyFractions]{-b/d}$ (we could need this in our final answer).

Now multiply both sides of

\\[\\simplify{(x+{a})/({d}x+{b})} \\var{latex(sym)} \\var{c}\\]

by $(\\simplify{{d}x+{b}})^2$. This will get rid of the denominator with the added benefit that we know this quantity must be positive (it can't be negative and it can't be zero!) so we don't need to worry about swapping the direction of the inequality.

\\[\\simplify{(x+{a})*({d}x+{b})} \\var{latex(sym)} \\simplify{{c}*({d}x+{b})^2}\\]

This introduces a quadratic and so we can continue this question using the same method that we use when dealing with inequalities involving quadratics (we also need to recall $x\\ne \\var[basic,fractionNumbers,simplifyFractions]{-b/d}$).

Let's get everything on the right hand side and factorise the common factor (our original denominator) out.

\\[\\begin{align}\\simplify{(x+{a})*({d}x+{b})} &\\var{latex(sym)} \\simplify{{c}*({d}x+{b})^2}\\\\ 0 &\\var{latex(sym)} \\simplify{{c}*({d}x+{b})^2-(x+{a})*({d}x+{b})} \\\\ 0 &\\var{latex(sym)}\\simplify{({d}x+{b})({c}({d}x+{b})-(x+{a}))}\\\\ 0&\\var{latex(sym)} \\simplify{({d}x+{b})({c*d-1}x+{c*b-a})} \\end{align}\\]

We are probably more used to seeing this written the other way

\\[\\simplify{({d}x+{b})({c*d-1}x+{c*b-a})}\\var{latex(backsym)} 0\\]

Since the quadratic is now factorised we can see the roots are $\\var[basic,fractionNumbers,simplifyFractions]{r1}$ and $\\var[basic,fractionNumbers,simplifyFractions]{r2}$.

Now we can either test the sign of each region $x< \\var[basic,fractionNumbers,simplifyFractions]{r1}$, $\\var[basic,fractionNumbers,simplifyFractions]{r1}<x<\\var[basic,fractionNumbers,simplifyFractions]{r2}$, and $x>\\var[basic,fractionNumbers,simplifyFractions]{r2}$ or we can sketch the parabola and determine which regions satisfy the inequality.

I'll use the parabola method:

The leading coefficient of the quadratic is negative positive and therefore the parabola is concave down up.
We want the quadratic to be greater than greater than or equal to less than less than or equal to zero and hence we want the parabola to be above above or on below below or on the $x$ axis. Except that $x\\ne \\var[basic,fractionNumbers,simplifyFractions]{-b/d}$.

So we have the following situation (where red indicates the inequality is not satisifed and green indicates it is):

{graph()}

Therefore the solution to the original inequality is

\\[\\var[basic,fractionNumbers,simplifyFractions]{r1} < x < \\var[basic,fractionNumbers,simplifyFractions]{r2}.\\]

\\[\\var[basic,fractionNumbers,simplifyFractions]{r1} < x \\le \\var[basic,fractionNumbers,simplifyFractions]{r2}.\\]

\\[\\var[basic,fractionNumbers,simplifyFractions]{r1} \\le x < \\var[basic,fractionNumbers,simplifyFractions]{r2}.\\]

\\[ x<\\var[basic,fractionNumbers,simplifyFractions]{r1},\\; x>\\var[basic,fractionNumbers,simplifyFractions]{r2}.\\]

\\[ x\\leq\\var[basic,fractionNumbers,simplifyFractions]{r1},\\; x>\\var[basic,fractionNumbers,simplifyFractions]{r2}.\\]

\\[ x<\\var[basic,fractionNumbers,simplifyFractions]{r1},\\; x\\ge\\var[basic,fractionNumbers,simplifyFractions]{r2}.\\]

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Recall that with inequalities, you need to know whether an expression is positive or negative before you multiply or divide by it.

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The solution to the inequality

\\[\\simplify{(x+{a})/({d}x+{b})} \\var{latex(sym)} \\var{c}\\]

is made up of [[0]]

Given by the condition [[1]] [[2]] $x$ [[3]] [[4]].

Given by the condition, $x$ [[5]] [[6]] or $x$ [[7]] [[8]].

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Harder inequalities that involve terms like $\\frac{1}{x}$ where multiplication by the denominator requires you to split into cases for positive or negative, or you multiply by the square of the denominator.

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