Solving linear inequalities

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Xiaodan Leng 5 years, 8 months ago

Created this as a copy of Solving linear inequalities.

Name	Status	Author	Last Modified
Solving linear inequalities	Ready to use	Bradley Bush	20/11/2019 14:42
Solve quadratic inequalities	Ready to use	Bradley Bush	20/11/2019 14:42
Solving linear inequalities	Should not be used	Adrian Jannetta	31/08/2023 17:00
Ida's copy of Solve quadratic inequalities	draft	Ida Landgärds	13/04/2018 09:16
Ida's copy of Solving linear inequalities	draft	Ida Landgärds	13/04/2018 09:16
Solving linear inequalities [L5 Randomised]	Needs to be tested	Matthew James Sykes	25/07/2018 10:51
cormac's copy of Solving linear inequalities	draft	cormac breen	24/10/2018 13:44
cormac's copy of Solve quadratic inequalities	draft	cormac breen	24/10/2018 13:46
Solving linear inequalities	draft	Xiaodan Leng	11/07/2019 01:43

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Question statement

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

Give any introductory information the student needs.

			Name	Type	Generated Value

a	list	List of 10 items
b	list	List of 10 items
c	integer	6
x		Definition of variable `x` is empty.

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Name

Data type

Value

xxxxxxxxxx
 
repeat(random(2..10),10)

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Description

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Generated value: `list`

[ 7, 9, 10, 2, 9, 5, 10, 5, 6, 10 ]

→ Used by:

Advice
"Part a)" - prompt
"Part a)" → "Unnamed gap" - Correct answer
"Part b)" - prompt
"Part b)" → "Unnamed gap" - Correct answer
"Part c)" - prompt
"Part c)" → "Unnamed gap" - Correct answer
"Part d)" - prompt
"Part d)" → "Gap 0." - Correct answer
"Part f)" - prompt
"Part f)" → "Gap 0." - Correct answer
"Part g)" - prompt
"Part g)" → "Gap 0." - Correct answer

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2. Gap Gap 1. Choose one from a list
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Gap-fill

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

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

$x<$ Unnamed gap

Advice

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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This question is used in the following exam:

Basics Algebra by Max S in Max's workspace.

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Xiaodan Leng 5 years, 8 months ago

Error in variable testing condition

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Generated value: `list`

Parts

Gap-fill

a)

b)

c)

d)

e)

f)

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Solving linear inequalities

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Xiaodan Leng 5 years, 8 months ago

Error in variable testing condition

Error

Warning

Generated value: list

Parts

Gap-fill

a)

b)

c)

d)

e)

f)

Generated value: `list`