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<!DOCTYPE html>
<html data-require="math math-format expressions graphie interactive">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Graphing and solving systems of inequalities</title>
<script data-main="../local-only/main.js" src="../local-only/require.js"></script>
</head>
<body>
<div class="exercise">
<div class="problems">
<div>
<div class="vars">
<div data-ensure="abs(YINT_1 + SLOPE_FRAC_1[0]) &lt;= 10 &amp;&amp; abs(YINT_2 + SLOPE_FRAC_2[0]) &lt;= 10">
<var id="X">randRangeNonZero(-5, 5)</var>
<var id="Y">randRange(-5, 5)</var>
<div data-ensure="abs(YINT_1 - YINT_2) &gt; 3">
<var id="YINT_1">randRangeExclude(-5, 5, [0, Y])</var>
<var id="YINT_2">randRangeExclude(-5, 5, [0, Y])</var>
</div>
<var id="SLOPE_1">(Y - YINT_1) / X</var>
<var id="SLOPE_2">(Y - YINT_2) / X</var>
<var id="SLOPE_FRAC_1">toFraction(SLOPE_1, 0.001)</var>
<var id="SLOPE_FRAC_2">toFraction(SLOPE_2, 0.001)</var>
</div>
<var id="PRETTY_SLOPE_1">fractionVariable(SLOPE_FRAC_1[0], SLOPE_FRAC_1[1], "x")</var>
<var id="PRETTY_SLOPE_2">fractionVariable(SLOPE_FRAC_2[0], SLOPE_FRAC_2[1], "x")</var>
<var id="MULT_1">randRangeNonZero(-3, 3)</var>
<var id="MULT_2">randRangeNonZero(-3, 3)</var>
<var id="A_1">SLOPE_FRAC_1[0] * -MULT_1</var>
<var id="A_2">SLOPE_FRAC_2[0] * -MULT_2</var>
<var id="B_1">SLOPE_FRAC_1[1] * MULT_1</var>
<var id="B_2">SLOPE_FRAC_2[1] * MULT_2</var>
<var id="C_1">SLOPE_FRAC_1[1] * YINT_1 * MULT_1</var>
<var id="C_2">SLOPE_FRAC_2[1] * YINT_2 * MULT_2</var>
<var id="STD_FORM_1">randFromArray([true, false])</var>
<var id="STD_FORM_2">randFromArray([true, false])</var>
<var id="COMP_1">randFromArray(["&lt;", "&gt;", "≤", "≥"])</var>
<var id="COMP_2">randFromArray(["&lt;", "&gt;", "≤", "≥"])</var>
<var id="STD_FORM_COMP_1">B_1 &lt; 0 ? { "&lt;": "&gt;", "&gt;": "&lt;", "≤": "≥", "≥": "≤" }[COMP_1] : COMP_1</var>
<var id="STD_FORM_COMP_2">B_2 &lt; 0 ? { "&lt;": "&gt;", "&gt;": "&lt;", "≤": "≥", "≥": "≤" }[COMP_2] : COMP_2</var>
<var id="LESS_THAN_1">COMP_1 === "&lt;" || COMP_1 === "≤"</var>
<var id="LESS_THAN_2">COMP_2 === "&lt;" || COMP_2 === "≤"</var>
<var id="INCLUSIVE_1">COMP_1 === "≥" || COMP_1 === "≤"</var>
<var id="INCLUSIVE_2">COMP_2 === "≥" || COMP_2 === "≤"</var>
<var id="EDGE_POINT" data-ensure="abs(EDGE_POINT[0]) < 10 && abs(EDGE_POINT[1]) < 10">(function() {
// Create a point on one of the lines
var minN = -floor((9 + X) / abs(SLOPE_FRAC_1[1]));
var maxN = floor((9 - X) / abs(SLOPE_FRAC_1[1]));
var n = randRange(minN, maxN);
if (rand(2) &lt; 1) {
return [X + n * SLOPE_FRAC_1[1], Y + n * SLOPE_FRAC_1[0]];
} else {
return [X + n * SLOPE_FRAC_2[1], Y + n * SLOPE_FRAC_2[0]];
}
})()</var>
<!-- !/3 of the time one of the points should be on the line -->
<var id="POINT">randFromArray([
[randRangeExclude(-9, 9, [-1, -2]), randRangeExclude(-9, 9, [-1, -2])],
[randRangeExclude(-9, 9, [-1, -2]), randRangeExclude(-9, 9, [-1, -2])],
EDGE_POINT
])
</var>
<var id="POINT_SOLUTION">(function() {
var p = POINT[1];
var p1 = SLOPE_1 * POINT[0] + YINT_1;
var p2 = SLOPE_2 * POINT[0] + YINT_2;
return ((COMP_1 === "&lt;" &amp;&amp; p &lt; p1) || (COMP_1 === "≤" &amp;&amp; p &lt;= p1) ||
(COMP_1 === "&gt;" &amp;&amp; p &gt; p1) || (COMP_1 === "≥" &amp;&amp; p &gt;= p1)) &amp;&amp;
((COMP_2 === "&lt;" &amp;&amp; p &lt; p2) || (COMP_2 === "≤" &amp;&amp; p &lt;= p2) ||
(COMP_2 === "&gt;" &amp;&amp; p &gt; p2) || (COMP_2 === "≥" &amp;&amp; p &gt;= p2));
})()</var>
</div>
<p class="question">
Graph the following system of inequalities:
</p>
<div class="problem">
<p data-if="STD_FORM_1">
<code>\blue{<var>expr([ "+", [ "*", A_1, "x" ], [ "*", B_1, "y" ] ])</var> <var>STD_FORM_COMP_1</var> <var>C_1</var>}</code>
</p><p data-else="">
<code>\blue{y <var>COMP_1</var> <var>PRETTY_SLOPE_1</var> + <var>YINT_1</var>}</code>
</p>
<p data-if="STD_FORM_2">
<code>\green{<var>expr([ "+", [ "*", A_2, "x" ], [ "*", B_2, "y" ] ])</var> <var>STD_FORM_COMP_2</var> <var>C_2</var>}</code>
</p>
<p data-else="">
<code>\green{y <var>COMP_2</var> <var>PRETTY_SLOPE_2</var> + <var>YINT_2</var>}</code>
</p>
<form>
<span class="hint_blue" style="width: 40px">Inequality 1:</span>
<input onclick="javascript:
KhanUtil.currentGraph.graph.shadetop1 = !KhanUtil.currentGraph.graph.shadetop1;
KhanUtil.currentGraph.graph.update();
" type="button" value="Shade other side">
<ul class="inequalities-one-line-radios">
<li>
<label class="hint_blue">
<input checked name="dashradio1" onclick="javascript:
KhanUtil.currentGraph.graph.dasharray1 = '';
KhanUtil.currentGraph.graph.update();
" type="radio" value="solid">
Solid line
</label>
</li>
<li>
<label class="hint_blue">
<input name="dashradio1" onclick="javascript:
KhanUtil.currentGraph.graph.dasharray1 = '- ';
KhanUtil.currentGraph.graph.update();
" type="radio" value="dashed">
Dashed line
</label>
</li>
</ul>
<br>
<span class="hint_green" style="width: 40px">Inequality 2:</span>
<input onclick="javascript:
KhanUtil.currentGraph.graph.shadetop2 = !KhanUtil.currentGraph.graph.shadetop2;
KhanUtil.currentGraph.graph.update();
" type="button" value="Shade other side">
<ul class="inequalities-one-line-radios">
<li>
<label class="hint_green">
<input checked name="dashradio2" onclick="javascript:
KhanUtil.currentGraph.graph.dasharray2 = '';
KhanUtil.currentGraph.graph.update();
" type="radio" value="solid">
Solid line
</label>
</li>
<li>
<label class="hint_green">
<input name="dashradio2" onclick="javascript:
KhanUtil.currentGraph.graph.dasharray2 = '- ';
KhanUtil.currentGraph.graph.update();
" type="radio" value="dashed">
Dashed line
</label>
</li>
</ul>
</form>
<div class="graphie inequalities-padding" id="grid">
graphInit({
range: 11,
scale: 20,
axisArrows: "&lt;-&gt;",
tickStep: 1,
labelStep: 1,
gridOpacity: 0.05,
axisOpacity: 0.2,
tickOpacity: 0.4,
labelOpacity: 0.5
});
label( [ 0, -11 ], "y", "below" );
label( [ 11, 0 ], "x", "right" );
addMouseLayer();
graph.pointA = addMovablePoint({
coord: [ -5, 5 ],
snapX: 0.5,
snapY: 0.5,
normalStyle: {
stroke: KhanUtil.BLUE,
fill: KhanUtil.BLUE
}
});
graph.pointB = addMovablePoint({
coord: [ 5, 5 ],
snapX: 0.5,
snapY: 0.5,
normalStyle: {
stroke: KhanUtil.BLUE,
fill: KhanUtil.BLUE
}
});
graph.pointC = addMovablePoint({
coord: [ -5, -5 ],
snapX: 0.5,
snapY: 0.5,
normalStyle: {
stroke: KhanUtil.BLUE,
fill: KhanUtil.BLUE
}
});
graph.pointD = addMovablePoint({
coord: [ 5, -5 ],
snapX: 0.5,
snapY: 0.5,
normalStyle: {
stroke: KhanUtil.BLUE,
fill: KhanUtil.BLUE
}
});
graph.set = raphael.set();
graph.update = function() {
graph.set.remove();
if ( abs( graph.pointB.coord[0] - graph.pointA.coord[0] ) &gt; 0.001 ) {
var slope = ( graph.pointB.coord[1] - graph.pointA.coord[1] ) / ( graph.pointB.coord[0] - graph.pointA.coord[0] );
var yint = slope * ( 0 - graph.pointA.coord[0] ) + graph.pointA.coord[1];
var shadeEdge = ( ( graph.pointA.coord[0] &lt; graph.pointB.coord[0] ) ? graph.shadetop1 : !graph.shadetop1 ) ? 11 : -11;
style({ stroke: BLUE, strokeWidth: 2, strokeDasharray: graph.dasharray1 }, function() {
graph.set.push( line( [ -11, -11 * slope + yint ], [ 11, 11 * slope + yint ] ) );
});
style({ fill: BLUE, stroke: null, opacity: KhanUtil.FILL_OPACITY }, function() {
graph.set.push( path([ [ 11, shadeEdge ], [ 11, 11 * slope + yint ], [ -11, -11 * slope + yint ], [ -11, shadeEdge ] ]) );
});
} else {
// vertical line
var x = graph.pointA.coord[0];
var shadeEdge = ( ( graph.pointB.coord[1] &lt; graph.pointA.coord[1] ) ? graph.shadetop1 : !graph.shadetop1 ) ? 11 : -11;
style({ stroke: BLUE, strokeWidth: 2, strokeDasharray: graph.dasharray1 }, function() {
graph.set.push( line( [ x, -11 ], [ x, 11 ] ) );
});
style({ fill: BLUE, stroke: null, opacity: KhanUtil.FILL_OPACITY }, function() {
graph.set.push( path([ [ x, -11 ], [ x, 11 ], [ shadeEdge, 11 ], [ shadeEdge, -11 ] ]) );
});
}
if ( abs( graph.pointD.coord[0] - graph.pointC.coord[0] ) &gt; 0.001 ) {
var slope = ( graph.pointD.coord[1] - graph.pointC.coord[1] ) / ( graph.pointD.coord[0] - graph.pointC.coord[0] );
var yint = slope * ( 0 - graph.pointC.coord[0] ) + graph.pointC.coord[1];
var shadeEdge = ( ( graph.pointC.coord[0] &lt; graph.pointD.coord[0] ) ? graph.shadetop2 : !graph.shadetop2 ) ? 11 : -11;
style({ stroke: GREEN, strokeWidth: 2, strokeDasharray: graph.dasharray2 }, function() {
graph.set.push( line( [ -11, -11 * slope + yint ], [ 11, 11 * slope + yint ] ) );
});
style({ fill: GREEN, stroke: null, opacity: KhanUtil.FILL_OPACITY }, function() {
graph.set.push( path([ [ 11, shadeEdge ], [ 11, 11 * slope + yint ], [ -11, -11 * slope + yint ], [ -11, shadeEdge ] ]) );
});
} else {
// vertical line
var x = graph.pointC.coord[0];
var shadeEdge = ( ( graph.pointD.coord[1] &lt; graph.pointC.coord[1] ) ? graph.shadetop2 : !graph.shadetop2 ) ? 11 : -11;
style({ stroke: GREEN, strokeWidth: 2, strokeDasharray: graph.dasharray2 }, function() {
graph.set.push( line( [ x, -11 ], [ x, 11 ] ) );
});
style({ fill: GREEN, stroke: null, opacity: KhanUtil.FILL_OPACITY }, function() {
graph.set.push( path([ [ x, -11 ], [ x, 11 ], [ shadeEdge, 11 ], [ shadeEdge, -11 ] ]) );
});
}
graph.set.toBack();
};
graph.showCorrect = function() {
graph.pointA.setCoord([ 0, YINT_1 ]);
graph.pointB.setCoord([ SLOPE_FRAC_1[1], YINT_1 + SLOPE_FRAC_1[0] ]);
graph.pointC.setCoord([ 0, YINT_2 ]);
graph.pointD.setCoord([ SLOPE_FRAC_2[1], YINT_2 + SLOPE_FRAC_2[0] ]);
graph.shadetop1 = graph.pointA.coord[0] &gt; graph.pointB.coord[0] ? LESS_THAN_1 : !LESS_THAN_1;
graph.shadetop2 = graph.pointC.coord[0] &gt; graph.pointD.coord[0] ? LESS_THAN_2 : !LESS_THAN_2;
if ( INCLUSIVE_1 ) {
graph.dasharray1 = '';
$( 'input[name=dashradio1][value=solid]' ).attr( 'checked', true );
} else {
graph.dasharray1 = '- ';
$( 'input[name=dashradio1][value=dashed]' ).attr( 'checked', true );
}
if ( INCLUSIVE_2 ) {
graph.dasharray2 = '';
$( 'input[name=dashradio2][value=solid]' ).attr( 'checked', true );
} else {
graph.dasharray2 = '- ';
$( 'input[name=dashradio2][value=dashed]' ).attr( 'checked', true );
}
graph.update();
};
// A and B can't be in the same place
graph.pointA.onMove = function( x, y ) {
if ( x != graph.pointB.coord[0] || y != graph.pointB.coord[1] ) {
graph.pointA.setCoord([ x, y ]);
graph.update();
return true;
} else {
return false;
}
}
graph.pointB.onMove = function( x, y ) {
if ( x != graph.pointA.coord[0] || y != graph.pointA.coord[1] ) {
graph.pointB.setCoord([ x, y, ]);
graph.update();
return true;
} else {
return false;
}
}
// C and D can't be in the same place
graph.pointC.onMove = function( x, y ) {
if ( x != graph.pointD.coord[0] || y != graph.pointD.coord[1] ) {
graph.pointC.setCoord([ x, y ]);
graph.update();
return true;
} else {
return false;
}
}
graph.pointD.onMove = function( x, y ) {
if ( x != graph.pointC.coord[0] || y != graph.pointC.coord[1] ) {
graph.pointD.setCoord([ x, y, ]);
graph.update();
return true;
} else {
return false;
}
}
graph.dasharray1 = "";
graph.dasharray2 = "";
graph.shadetop1 = true;
graph.shadetop2 = false;
graph.update();
graph.pointA.toFront();
graph.pointB.toFront();
graph.pointC.toFront();
graph.pointD.toFront();
</div>
<p>
Is <code>(<var>POINT[0]</var>, <var>POINT[1]</var>)</code> a solution of the inequality?
</p>
<div class="render-answer-area-here"></div>
</div>
<div class="solution" data-type="multiple">
<div class="instruction">
<ul>
<li>
<label>
<input id="yes" checked name="isSolution" type="radio">
<span>Yes</span>
</label>
</li>
<li>
<label>
<input id="no" name="isSolution" type="radio">
<span>No</span>
</label>
</li>
</ul>
</div>
<div class="sol" data-type="custom">
<div class="instruction"></div>
<div class="guess">[
graph.pointA.coord, graph.pointB.coord,
graph.pointA.coord[0] &gt; graph.pointB.coord[0] ? graph.shadetop1 : !graph.shadetop1,
graph.dasharray1 === "- " ? false : true,
graph.pointC.coord, graph.pointD.coord,
graph.pointC.coord[0] &gt; graph.pointD.coord[0] ? graph.shadetop2 : !graph.shadetop2,
graph.dasharray2 === "- " ? false : true,
$("input[name='isSolution']:checked").attr("id")]
</div>
<div class="validator-function">
if (_.isEqual(guess, [[-5,5],[5,5],false,true,[-5,-5],[5,-5],true,true])) {
return "";
}
var slope1 = ( guess[1][1] - guess[0][1] ) / ( guess[1][0] - guess[0][0] );
var yint1 = slope1 * ( 0 - guess[0][0] ) + guess[0][1];
var slope2 = ( guess[5][1] - guess[4][1] ) / ( guess[5][0] - guess[4][0] );
var yint2 = slope2 * ( 0 - guess[4][0] ) + guess[4][1];
return POINT_SOLUTION === (guess[8] === 'yes') &amp;&amp;
(abs(SLOPE_1 - slope1) &lt; 0.001
&amp;&amp; abs(YINT_1 - yint1) &lt; 0.001
&amp;&amp; guess[2] === LESS_THAN_1
&amp;&amp; guess[3] === INCLUSIVE_1
&amp;&amp; abs(SLOPE_2 - slope2) &lt; 0.001
&amp;&amp; abs(YINT_2 - yint2) &lt; 0.001
&amp;&amp; guess[6] === LESS_THAN_2
&amp;&amp; guess[7] === INCLUSIVE_2)
|| (abs(SLOPE_2 - slope1) &lt; 0.001
&amp;&amp; abs(YINT_2 - yint1) &lt; 0.001
&amp;&amp; guess[2] === LESS_THAN_2
&amp;&amp; guess[3] === INCLUSIVE_2
&amp;&amp; abs(SLOPE_1 - slope2) &lt; 0.001
&amp;&amp; abs(YINT_1 - yint2) &lt; 0.001
&amp;&amp; guess[6] === LESS_THAN_1
&amp;&amp; guess[7] === INCLUSIVE_1)
</div>
<div class="show-guess">
graph.pointA.setCoord( guess[0] );
graph.pointB.setCoord( guess[1] );
graph.pointC.setCoord( guess[4] );
graph.pointD.setCoord( guess[5] );
graph.shadetop1 = graph.pointA.coord[0] &gt; graph.pointB.coord[0] ? guess[2] : !guess[2];
graph.shadetop2 = graph.pointC.coord[0] &gt; graph.pointD.coord[0] ? guess[6] : !guess[6];
if ( guess[3] ) {
graph.dasharray1 = "";
$( "input[name=dashradio1][value=solid]" ).attr( "checked", true );
} else {
graph.dasharray1 = "- ";
$( "input[name=dashradio1][value=dashed]" ).attr( "checked", true );
}
if ( guess[7] ) {
graph.dasharray2 = "";
$( "input[name=dashradio2][value=solid]" ).attr( "checked", true );
} else {
graph.dasharray2 = "- ";
$( "input[name=dashradio2][value=dashed]" ).attr( "checked", true );
}
graph.update();
</div>
</div>
</div>
</div>
</div>
<div class="hints">
<div>
<p>
Let's first graph the boundary lines.
If an inequality is in slope-intercept form, we can use it to
determine the slope and the <code>y</code>-intercept of the line.
If an inequality isn't in slope-intercept form, let's convert it to this form.
</p>
<p>
The inequality signs in each inequality then tell us which side of
the lines are to be shaded and whether the lines are solid or dashed.
</p>
<p>
Finally, we can see whether or not our given point lies in the shaded area
representing both inequalities to check if it's a solution of the system.
</p>
</div>
<div>
<div data-if="STD_FORM_1">
<p>
The first inequality,
<code>\blue{<var>expr(["+", ["*", A_1, "x"], ["*", B_1, "y"]])</var><var>STD_FORM_COMP_1</var><var>C_1</var>}</code>,
isn't in slope-intercept form. Let's convert it:
</p>
<p><code>
\qquad\begin{eqnarray}
<var>expr(["+", ["*", A_1, "x"], ["*", B_1, "y"]])</var> &amp;<var>STD_FORM_COMP_1</var>&amp; <var>C_1</var> \\
<var>coefficient(B_1)</var>y &amp;<var>STD_FORM_COMP_1</var>&amp; <var>expr([ "+", [ "*", -A_1, "x" ], C_1 ])</var> \\
y &amp;<var>COMP_1</var>&amp; <var>PRETTY_SLOPE_1</var> + <var>YINT_1</var>
\end{eqnarray}
</code></p>
<p>
Now we see the slope of the line is
<code>\pink{<var>fractionReduce(SLOPE_FRAC_1[0], SLOPE_FRAC_1[1])</var>}</code>
and its <code>y</code>-intercept is <code>\purple{(0, <var>YINT_1</var>)}.</code>
</p>
</div>
<p data-else="">
The first inequality, <code>\blue{y <var>COMP_1</var> <var>PRETTY_SLOPE_1</var> + <var>YINT_1</var>}</code>,
is in slope-intercept form: the slope of the line is
<code><var>fractionReduce(SLOPE_FRAC_1[0], SLOPE_FRAC_1[1])</var></code>
and its <code>y</code>-intercept is <code>(0, <var>YINT_1</var>).</code>
</p>
<p>
According to the slope, we know the line also passes through
<code>(0 \pink{+ <var>SLOPE_FRAC_1[1]</var>}, <var>YINT_1</var> \purple{+ <var>SLOPE_FRAC_1[0]</var>})
= (<var>SLOPE_FRAC_1[1]</var>, <var>YINT_1 + SLOPE_FRAC_1[0]</var>)
</code>.
</p>
<div class="graphie" data-update="grid">
graph.pointA.setCoord([0, YINT_1]);
graph.pointB.setCoord([SLOPE_FRAC_1[1], YINT_1 + SLOPE_FRAC_1[0]]);
graph.update();
</div>
</div>
<div>
<div data-if="STD_FORM_2">
<p>
The second inequality,
<code>\green{<var>expr(["+", ["*", A_2, "x"], ["*", B_2, "y"]])</var><var>STD_FORM_COMP_2</var><var>C_2</var>}</code>,
isn't in slope-intercept form. Let's convert it:
</p>
<p><code>
\qquad\begin{eqnarray}
<var>expr(["+", ["*", A_2, "x"], ["*", B_2, "y"]])</var> &amp;<var>STD_FORM_COMP_2</var>&amp; <var>C_2</var> \\
<var>coefficient(B_2)</var>y &amp;<var>STD_FORM_COMP_2</var>&amp; <var>expr([ "+", [ "*", -A_2, "x" ], C_2 ])</var> \\
y &amp;<var>COMP_2</var>&amp; <var>PRETTY_SLOPE_2</var> + <var>YINT_2</var>
\end{eqnarray}
</code></p>
<p>
Now we see the slope of the line is
<code>\pink{<var>fractionReduce(SLOPE_FRAC_2[0], SLOPE_FRAC_2[1])</var>}</code>
and its <code>y</code>-intercept is <code>\purple{(0, <var>YINT_2</var>)}.</code>
</p>
</div>
<p data-else="">
The second inequality, <code>\green{y <var>COMP_2</var> <var>PRETTY_SLOPE_2</var> + <var>YINT_2</var>}</code>,
is in slope-intercept form: the slope of the line is
<code><var>fractionReduce(SLOPE_FRAC_2[0], SLOPE_FRAC_2[1])</var></code>
and its <code>y</code>-intercept is <code>(0, <var>YINT_2</var>).</code>
</p>
<p>
According to the slope, we know the line also passes through
<code>(0 \pink{+ <var>SLOPE_FRAC_2[1]</var>}, <var>YINT_2</var> \purple{+ <var>SLOPE_FRAC_2[0]</var>})
= (<var>SLOPE_FRAC_2[1]</var>, <var>YINT_2 + SLOPE_FRAC_2[0]</var>)
</code>.
</p>
<div class="graphie" data-update="grid">
graph.pointC.setCoord([0, YINT_2]);
graph.pointD.setCoord([SLOPE_FRAC_2[1], YINT_2 + SLOPE_FRAC_2[0]]);
graph.update();
</div>
</div>
<div>
<div data-if="LESS_THAN_1">
<p data-if="INCLUSIVE_1">
According to the sign of the first inequality,
<code>\blue{y <var>COMP_1</var> <var>PRETTY_SLOPE_1</var> + <var>YINT_1</var>}</code>,
its solution set lies <em>below</em> the boundary line and the line should be <em>solid</em>.
</p><p data-else="">
According to the sign of the first inequality,
<code>\blue{y <var>COMP_1</var> <var>PRETTY_SLOPE_1</var> + <var>YINT_1</var>}</code>,
its solution set lies <em>below</em> the boundary line and the line should be <em>dashed</em>.
</p>
</div><div data-else="">
<p data-if="INCLUSIVE_1">
According to the sign of the first inequality,
<code>\blue{y <var>COMP_1</var> <var>PRETTY_SLOPE_1</var> + <var>YINT_1</var>}</code>,
its solution set lies <em>above</em> the boundary line and the line should be <em>solid</em>.
</p><p data-else="">
According to the sign of the first inequality,
<code>\blue{y <var>COMP_1</var> <var>PRETTY_SLOPE_1</var> + <var>YINT_1</var>}</code>,
its solution set lies <em>above</em> the boundary line and the line should be <em>dashed</em>.
</p>
</div>
<div data-if="LESS_THAN_2">
<p data-if="INCLUSIVE_2">
According to the sign of the second inequality,
<code>\green{y <var>COMP_2</var> <var>PRETTY_SLOPE_2</var> + <var>YINT_2</var>}</code>,
its solution set lies <em>below</em> the boundary line and the line should be <em>solid</em>.
</p><p data-else="">
According to the sign of the second inequality,
<code>\green{y <var>COMP_2</var> <var>PRETTY_SLOPE_2</var> + <var>YINT_2</var>}</code>,
its solution set lies <em>below</em> the boundary line and the line should be <em>dashed</em>.
</p>
</div><div data-else="">
<p data-if="INCLUSIVE_2">
According to the sign of the second inequality,
<code>\green{y <var>COMP_2</var> <var>PRETTY_SLOPE_2</var> + <var>YINT_2</var>}</code>,
its solution set lies <em>above</em> the boundary line and the line should be <em>solid</em>.
</p><p data-else="">
According to the sign of the second inequality,
<code>\green{y <var>COMP_2</var> <var>PRETTY_SLOPE_2</var> + <var>YINT_2</var>}</code>,
its solution set lies <em>above</em> the boundary line and the line should be <em>dashed</em>.
</p>
</div>
<div class="graphie" data-update="grid">
graph.shadetop1 = graph.pointA.coord[0] &gt; graph.pointB.coord[0] ? LESS_THAN_1 : !LESS_THAN_1;
graph.shadetop2 = graph.pointC.coord[0] &gt; graph.pointD.coord[0] ? LESS_THAN_2 : !LESS_THAN_2;
if (INCLUSIVE_1) {
graph.dasharray1 = '';
$('input[name=dashradio1][value=solid]').attr('checked', true);
} else {
graph.dasharray1 = '- ';
$('input[name=dashradio1][value=dashed]').attr('checked', true);
}
if (INCLUSIVE_2) {
graph.dasharray2 = '';
$('input[name=dashradio2][value=solid]').attr('checked', true);
} else {
graph.dasharray2 = '- ';
$('input[name=dashradio2][value=dashed]').attr('checked', true);
}
graph.update();
</div>
</div>
<div>
<div class="graphie" data-update="grid">
style({ stroke: RED, fill: RED }, function() {
circle(POINT, 0.2);
label(POINT, "\\red{(" + POINT[0] + ", " + POINT[1] + ")}", "right");
});
</div>
<p data-if="POINT_SOLUTION">
We can see that the point
<code>\red{(<var>POINT[0]</var>, <var>POINT[1]</var>)}</code>
lies in the shaded area representing <em>both</em> inequalities,
which means that it <em>is</em> a solution of the system represented by the graph.
</p><p data-else="">
We can see that the point
<code>\red{(<var>POINT[0]</var>, <var>POINT[1]</var>)}</code>
does not lie in the shaded area representing <em>both</em> inequalities,
which means that it <em>is not</em> a solution of the system represented by the graph.
</p>
</div>
</div>
</div>
</body>
</html>
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