Geometric and Algebraic Demonstration of |x| > a
2020-03-10 00:04
This GeoGebra applet demonstrates the solution and geometric meaning of the absolute value inequality |x| > a. It shows the dynamic solution set (x > a or x < -a) as parameter a varies. Using Segment and If commands, it illustrates the interval representation on the real number line, helping students grasp the distance concept of absolute value and inequality solving logic.
In collections
Secondary Vocational Mathematics
Loading the GeoGebra board and drawing, please wait…
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.574.0" app="classic" platform="w" id="f1cfa941-8242-43a7-a55f-40730eb5b0b0" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/apps/xsd/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="1350" height="541" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.14814814814814814" orientation="1" />
</panes>
<views>
<view id="512" toolbar="0 | 1 501 5 19 , 67 | 2 15 45 18 , 7 37 | 514 3 9 , 13 44 , 47 | 16 51 | 551 550 11 , 20 22 21 23 , 55 56 57 , 12 | 69 | 510 511 , 512 513 | 533 531 , 534 532 , 522 523 , 537 536 , 535 | 521 520 | 36 , 38 49 560 | 571 30 29 570 31 33 | 17 | 540 40 41 42 , 27 28 35 , 6 , 502" visible="false" inframe="false" stylebar="false" location="1,1,1" size="500" window="100,100,600,400" />
<view id="4" toolbar="0 || 2020 , 2021 , 2022 , 66 || 2001 , 2003 , 2002 , 2004 , 2005 || 2040 , 2041 , 2042 , 2044 , 2043" visible="false" inframe="false" stylebar="false" location="1,1" size="300" window="100,100,600,400" />
<view id="8" toolbar="1001 | 1002 | 1003 || 1005 | 1004 || 1006 | 1007 | 1010 || 1008 1009 || 66 68 || 6" visible="false" inframe="false" stylebar="false" location="1,3" size="300" window="100,100,600,400" />
<view id="4097" visible="false" inframe="false" stylebar="true" location="1,1" size="499" window="100,100,700,550" />
<view id="1" visible="true" inframe="false" stylebar="false" location="1" size="1142" window="100,100,600,400" />
<view id="2" visible="true" inframe="false" stylebar="false" location="3" size="200" window="100,100,600,400" />
<view id="16" visible="false" inframe="false" stylebar="false" location="1" size="150" window="50,50,500,500" />
<view id="32" visible="false" inframe="false" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="64" toolbar="0" visible="false" inframe="false" stylebar="false" location="1" size="150" window="50,50,500,500" />
<view id="128" visible="false" inframe="false" stylebar="false" location="1" size="480" window="50,50,500,500" />
<view id="70" toolbar="0 || 2020 || 2021 || 2022" visible="false" inframe="false" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="43" visible="false" inframe="false" stylebar="false" location="1" size="450" window="50,50,500,500" />
</views>
<toolbar show="false" items="0 39 73 62 | 1 501 67 , 5 19 , 72 75 76 | 2 15 45 , 18 65 , 7 37 | 4 3 8 9 , 13 44 , 58 , 47 | 16 51 64 , 70 | 10 34 53 11 , 24 20 22 , 21 23 | 55 56 57 , 12 | 36 46 , 38 49 50 , 71 14 68 | 30 29 54 32 31 33 | 25 17 26 60 52 61 | 40 41 42 , 27 28 35 , 6" position="1" help="false" />
<input show="false" cmd="true" top="algebra" />
<dockBar show="false" east="false" />
</perspective>
</perspectives>
<labelingStyle val="0"/>
<font size="20"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="1142" height="541"/>
<coordSystem xZero="414.352385199363" yZero="296.7814466722058" scale="47.2836373136856" yscale="47.28363731368566"/>
<evSettings axes="true" grid="false" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" checkboxSize="26" gridType="3"/>
<bgColor r="255" g="255" b="255"/>
<axesColor r="0" g="0" b="0"/>
<gridColor r="192" g="192" b="192"/>
<lineStyle axes="1" grid="0"/>
<axis id="0" show="true" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
<axis id="1" show="false" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
</euclidianView>
<algebraView>
<mode val="3"/>
</algebraView>
<kernel>
<continuous val="false"/>
<usePathAndRegionParameters val="true"/>
<decimals val="2"/>
<angleUnit val="degree"/>
<algebraStyle val="0" spreadsheet="0"/>
<coordStyle val="0"/>
</kernel>
<tableview min="-2" max="2" step="1"/>
<scripting blocked="false" disabled="false"/>
<construction title="" author="" date="">
<element type="numeric" label="a">
<value val="2"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="-5" max="5" absoluteScreenLocation="true" width="200" x="276" y="59" fixed="false" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="1" speed="1" type="0" playing="false"/>
<ggbscript onUpdate="b=If(a>0,true,false)
c=If(a<0,true,false)
d=If(a>=0,true,false)
jieji=If(a==0,"解集为：{x|x≠0}",If(a>0,"解集为：{x|x<-"a"或x>"a"}",If(a<0,"解集为：R")))
"/>
</element>
<expression label="A" exp="(a, 0)" type="point" />
<element type="point" label="A">
<show object="true" label="false" ev="4"/>
<condition showObject="d"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="2" y="0" z="1"/>
<pointSize val="5"/>
<pointStyle val="2"/>
</element>
<expression label="B" exp="((-a), 0)" type="point" />
<element type="point" label="B">
<show object="true" label="false" ev="4"/>
<condition showObject="d"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-2" y="0" z="1"/>
<pointSize val="5"/>
<pointStyle val="2"/>
</element>
<command name="Segment">
<input a0="((-abs(a)) - 10, 0.3)" a1="((-abs(a)), 0.3)"/>
<output a0="f"/>
</command>
<element type="segment" label="f">
<show object="true" label="false" ev="4"/>
<condition showObject="b"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="true"/>
<coords x="0" y="10" z="-2.9999999999999996"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="(abs(a) + 10, 0.3)" a1="(abs(a), 0.3)"/>
<output a0="g"/>
</command>
<element type="segment" label="g">
<show object="true" label="false" ev="4"/>
<condition showObject="b"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="true"/>
<coords x="0" y="-10" z="2.9999999999999996"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="((-abs(a)), 0.3)" a1="((-abs(a)), 0)"/>
<output a0="h"/>
</command>
<element type="segment" label="h">
<show object="true" label="false" ev="4"/>
<condition showObject="d"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="true"/>
<coords x="0.3" y="0" z="0.6"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="(abs(a), 0.3)" a1="(abs(a), 0)"/>
<output a0="i"/>
</command>
<element type="segment" label="i">
<show object="true" label="false" ev="4"/>
<condition showObject="d"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="true"/>
<coords x="0.3" y="0" z="-0.6"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<expression label="bds" exp=""|x|>" + (LaTeX[a]) + """ />
<element type="text" label="bds">
<show object="true" label="true" ev="8"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="2" size="20" style="3"/>
<auxiliary val="true"/>
<absoluteScreenLocation x="268" y="95"/>
</element>
<command name="Segment">
<input a0="(-10, 0.3)" a1="(10, 0.3)"/>
<output a0="实数集R"/>
</command>
<element type="segment" label="实数集R">
<show object="true" label="false" ev="4"/>
<condition showObject="c"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="true"/>
<coords x="0" y="20" z="-6"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="If">
<input a0="a > 0" a1="true" a2="false"/>
<output a0="b"/>
</command>
<element type="boolean" label="b">
<value val="true"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="true"/>
<checkbox fixed="true"/>
</element>
<command name="If">
<input a0="a < 0" a1="true" a2="false"/>
<output a0="c"/>
</command>
<element type="boolean" label="c">
<value val="false"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="true"/>
<checkbox fixed="true"/>
</element>
<command name="If">
<input a0="a ≥ 0" a1="true" a2="false"/>
<output a0="d"/>
</command>
<element type="boolean" label="d">
<value val="true"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="true"/>
<checkbox fixed="true"/>
</element>
<command name="If">
<input a0="a ≟ 0" a1=""解集为：{x|x≠0}"" a2="If[a > 0, (((("解集为：{x|x<-" * a) * "或x>") * a) * "}"), If[a < 0, "解集为：R"]]"/>
<output a0="jieji"/>
</command>
<element type="text" label="jieji">
<show object="true" label="true" ev="8"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<font serif="true" sizeM="2" size="20" style="3"/>
<absoluteScreenLocation x="269" y="199"/>
</element>
</construction>
</geogebra>