Quadratic Inequalities

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Ben Brawn

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Ben Brawn

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7 years, 10 months ago

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Ben Brawn 7 years, 10 months ago

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Ben Brawn 7 years, 10 months ago

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Ben Brawn 7 years, 10 months ago

Created this as a copy of Domain of a square root function.

Name	Status	Author	Last Modified
Domain of a polynomial	Ready to use	Ben Brawn	19/10/2023 02:28
Domain of a rational function	Ready to use	Ben Brawn	09/10/2023 11:15
Domain of an exponential	Ready to use	Ben Brawn	08/04/2025 10:05
Domain of a log function	Ready to use	Ben Brawn	09/10/2023 11:17
Domain of a piecewise function	Ready to use	Ben Brawn	12/10/2023 02:17
Domain of a square root function	Ready to use	Ben Brawn	04/10/2024 05:33
Domain of a trigonometric function (radians)	Has some problems	Ben Brawn	18/10/2024 00:24
Domain of a trigonometric function (degrees)	Ready to use	Ben Brawn	16/06/2017 06:05
Quadratic Inequalities	Ready to use	Ben Brawn	22/06/2017 07:24
Quadratic Inequalities 2	Ready to use	Ben Brawn	22/06/2017 07:17
Luis's copy of Quadratic Inequalities 2	draft	Luis Hernandez	17/01/2019 19:16
Kaye's copy of Quadratic Inequalities	Ready to use	Kaye Guan	18/01/2019 00:26
Joshua's copy of Domain of a piecewise function	draft	Joshua Calcutt	02/04/2019 02:05
Function Quiz Q5	draft	Joshua Calcutt	02/04/2019 07:31
Terry's copy of Quadratic Inequalities	draft	Terry Young	03/04/2019 17:33
Terry's copy of Quadratic Inequalities 2	draft	Terry Young	03/04/2019 17:34
Function Quiz Q5 function defined by { #2 (three parts)	draft	Joshua Calcutt	24/07/2020 07:38
Maria's copy of Function Quiz Q5	draft	Maria Aneiros	25/05/2019 07:25
Maria's copy of Domain of a square root function	Needs to be tested	Maria Aneiros	05/02/2020 06:54
Maria's copy of Domain of a rational function	Needs to be tested	Maria Aneiros	05/02/2020 06:50
Maria's copy of Domain of a polynomial	Needs to be tested	Maria Aneiros	05/02/2020 06:47
Maria's copy of Domain of a trigonometric function (degrees)	draft	Maria Aneiros	25/05/2019 07:58
Maria's copy of Domain of a trigonometric function (radians)	draft	Maria Aneiros	25/05/2019 08:00
Maria's copy of Domain of an exponential	draft	Maria Aneiros	25/05/2019 08:01
Maria's copy of Domain of a log function	draft	Maria Aneiros	05/02/2020 07:14
Quadratic Inequalities	draft	Xiaodan Leng	10/07/2019 22:39
Quadratic Inequalities	draft	Xiaodan Leng	11/07/2019 02:19
Quadratic Inequalities 2	draft	Xiaodan Leng	11/07/2019 02:19
Domain of a trigonometric function (degrees)	draft	Xiaodan Leng	11/07/2019 02:21
Domain of a trigonometric function (radians)	draft	Xiaodan Leng	11/07/2019 02:22
Domain of a square root function	draft	Xiaodan Leng	11/07/2019 02:22
Domain of a log function	draft	Xiaodan Leng	11/07/2019 02:23
Domain of a rational function	draft	Xiaodan Leng	11/07/2019 02:24
Function defined by { #1	draft	Yuri Anissimov	17/07/2020 01:20
Function defined by { #2	draft	Yuri Anissimov	20/07/2020 07:07
NC Math 3 Joshua's copy of Domain of a piecewise function	draft	Terry Young	10/02/2020 19:28
Definisjonsmengden piecewise	Ready to use	Elena Malyutina	02/09/2021 11:25
Definisjonsmengden polynom	Ready to use	Elena Malyutina	26/08/2024 14:48
Ugur's copy of Domain of a rational function	Ready to use	Ugur Efem	14/01/2022 15:06
Domain of a rational function short version	Ready to use	Ugur Efem	12/01/2022 11:44

There are 73 other versions that do you not have access to.

Question statement

The following will step you through solving the quadratic inequality

$\simplify{{a}x^2-{a}({r1}+{r2})x+{a*r1*r2}}\var{latex(sym)}0$

Give any introductory information the student needs.

			Name	Type	Generated Value

arootlist	list	[ -6, 7 ]
r1	number	-6
r2	number	7
a	integer	-1
choice	integer	3
sym	string	<
prod	integer	-3

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
sort(shuffle(-12..12)[0..2])

JME function reference

Description

Describe what this variable represents, and list any assumptions made about its value.

Can an exam override the value of this variable?

Generated value: `list`

[ -6, 7 ]

This variable doesn't seem to be used anywhere.

Parts

Part a) Gap-fill
1. Gap Gap 0. Number entry
  Add
2. Gap Gap 1. Number entry
  Add
Part b) Gap-fill
Part c) Gap-fill
1. Gap Choose one from a list
  Add
Part d) Choose one from a list

Gap-fill

Ask the student a question, and give any hints about how they should answer this part.

Find the roots of the quadratic involved in the inequality and order them in ascending order.

The smaller root is Gap 0. and the larger root is Gap 1..

Advice

The region approach is a common approach to solving an inequality that involves a quadratic:

The inequality $\simplify{{a}x^2-{a}({r1}+{r2})x+{a*r1*r2}}\var{latex(sym)}0$ can be written in a factorised form as $\simplify{({a}x-{a*r1})(x-{r2})}\var{latex(sym)}0$ which means the roots are $x=\var{r1},\, \var{r2}$.

The three regions are $x<\var{r1}$, $\var{r1}<x<\var{r2}$, and $x>\var{r2}$.

To test each region you can either choose a number in each region and actually check if the inequality is satisfied there (not recommended) or simply determine the sign, not the value, of each factor of the left hand side, that is:

When $x<\var{r1}$, we have that $(\simplify{({a}x-{a*r1})})$ is positive negative and $(\simplify{(x-{r2})})$ is negative and therefore $\simplify{({a}x-{a*r1})(x-{r2})}$ is positive negative. In otherwords, the inequality $\simplify{({a}x-{a*r1})(x-{r2})}\var{latex(sym)}0$ is satisfied not satisifed when $x<\var{r1}$.
When $\var{r1}<x<\var{r2}$, we have that $(\simplify{({a}x-{a*r1})})$ is positive negative and $(\simplify{(x-{r2})})$ is negative and therefore $\simplify{({a}x-{a*r1})(x-{r2})}$ is positive negative. In otherwords, the inequality $\simplify{({a}x-{a*r1})(x-{r2})}\var{latex(sym)}0$ is not satisfied satisifed when $\var{r1}<x<\var{r2}$.
When $x>\var{r2}$, we have that $(\simplify{({a}x-{a*r1})})$ is positive negative and $(\simplify{(x-{r2})})$ is positive and therefore $\simplify{({a}x-{a*r1})(x-{r2})}$ is positive negative. In otherwords, the inequality $\simplify{({a}x-{a*r1})(x-{r2})}\var{latex(sym)}0$ is satisfied not satisifed when $x>\var{r2}$.

At the roots, the quadratic will equal zero and hence the given inequality will not be satisfied since the inequality symbol $\var{latex(sym)}$ is not strict. That is, $0\not > 0$ $0 \ge 0$ $0\not < 0$ $0\le 0$.

Finally we can say the solution to the inequality is $x\le\var{r1}$, $x\ge\var{r2}$ $x<\var{r1}$, $x>\var{r2}$ $\var{r1} < x < \var{r2}$ $\var{r1} \le x \le \var{r2}$.

Another common approach is sketching the parabola $y=\simplify{{a}x^2-{a}({r1}+{r2})x+{a*r1*r2}}$ (by finding the roots and by noting the sign of the leading term determine if it is concave up or down) and then simply determining which parts of the graph satisfy the inequality. In the graph below, the green parts are those that correspond to $x$ values which satisfy the inequality and the red parts are those that do not.

{graph()}

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This question is used in the following exams:

Inequalities involving quadratics or hyperbolas by Ben Brawn in Algebra.
Algebra by James McNaughton in James's workspace.
Algebra by Dean Ang in Dean's workspace.
Algebra by Chris Templet in Chris's workspace.

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Ben Brawn 7 years, 10 months ago

Ben Brawn 7 years, 10 months ago

Ben Brawn 7 years, 10 months ago

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Quadratic Inequalities

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Ben Brawn 7 years, 10 months ago

Ben Brawn 7 years, 10 months ago

Ben Brawn 7 years, 10 months ago

Error in variable testing condition

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Warning

Generated value: list

Parts

Gap-fill

Generated value: `list`