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graphing_inequalities_2.html
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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 inequalities 2</title>
<script src="../khan-exercise.js"></script>
</head>
<body>
<div class="exercise">
<div class="problems">
<div>
<div class="vars">
<var id="SLOPE_FRAC">reduce( randRangeNonZero( -5, 5 ), randRange( 1, 5 ) )</var>
<var id="YINT">randRangeNonZero( max( -10, -10 - SLOPE_FRAC[0] ), min( 10, 10 - SLOPE_FRAC[0] ) )</var>
<var id="SLOPE">SLOPE_FRAC[0] / SLOPE_FRAC[1]</var>
<var id="PRETTY_SLOPE">SLOPE === 1 ? "" : ( SLOPE === -1 ? "-" : fraction( SLOPE_FRAC[0], SLOPE_FRAC[1], true, true ) )</var>
<var id="MULT">randRangeNonZero( -3, 3 )</var>
<var id="A">SLOPE_FRAC[0] * -MULT</var>
<var id="B">SLOPE_FRAC[1] * MULT</var>
<var id="C">SLOPE_FRAC[1] * YINT * MULT</var>
<var id="STD_FORM">randFromArray([ true, false ])</var>
<var id="COMP">randFromArray([ "<", ">", "≤", "≥" ])</var>
<var id="STD_FORM_COMP">B < 0 ? { "<": ">", ">": "<", "≤": "≥", "≥": "≤" }[ COMP ] : COMP</var>
<var id="LESS_THAN">COMP === "<" || COMP === "≤"</var>
<var id="INCLUSIVE">COMP === "≥" || COMP === "≤"</var>
<div data-ensure="sqrt( pow( POINT_1[0] - POINT_2[0], 2 ) + pow( POINT_1[1] - POINT_2[1], 2 ) ) > 4">
<var id="POINT_1">[ randRangeExclude( -9, 9, [ -3, -2, -1, 0 ] ), randRangeExclude( -9, 9, [ -1, -2 ] ) ]</var>
<var id="POINT_2">[ randRangeExclude( -9, 9, [ -3, -2, -1, 0 ] ), randRangeExclude( -9, 9, [ -1, -2 ] ) ]</var>
<var id="POINT_1_SOLUTION">
(((COMP === "<") && ( POINT_1[1] < SLOPE * POINT_1[0] + YINT ))
|| ((COMP === "≤") && ( POINT_1[1] <= SLOPE * POINT_1[0] + YINT ))
|| ((COMP === ">") && ( POINT_1[1] > SLOPE * POINT_1[0] + YINT ))
|| ((COMP === "≥") && ( POINT_1[1] >= SLOPE * POINT_1[0] + YINT )))
</var>
<var id="POINT_2_SOLUTION">
(((COMP === "<") && ( POINT_2[1] < SLOPE * POINT_2[0] + YINT ))
|| ((COMP === "≤") && ( POINT_2[1] <= SLOPE * POINT_2[0] + YINT ))
|| ((COMP === ">") && ( POINT_2[1] > SLOPE * POINT_2[0] + YINT ))
|| ((COMP === "≥") && ( POINT_2[1] >= SLOPE * POINT_2[0] + YINT )))
</var>
</div>
</div>
<p class="question">
Graph the following inequality and determine if the points
<code>(<var>POINT_1[0]</var>, <var>POINT_1[1]</var>)</code> or
<code>(<var>POINT_2[0]</var>, <var>POINT_2[1]</var>)</code> are solutions:
</p>
<div class="problem">
<p class="hint_blue" data-if="STD_FORM">
<code><var>expr([ "+", [ "*", A, "x" ], [ "*", B, "y" ] ])</var> <var>STD_FORM_COMP</var> <var>C</var></code>
</p>
<p class="hint_blue" data-else>
<code>y <var>COMP</var> <var>PRETTY_SLOPE</var> x + <var>YINT</var></code>
</p>
<form>
<input type="button" value="Shade other side" onClick="javascript:
KhanUtil.currentGraph.graph.shadetop = !KhanUtil.currentGraph.graph.shadetop;
KhanUtil.currentGraph.graph.update();
" />
<label>
<input name="dashradio" type="radio" value="solid" checked onClick="javascript:
KhanUtil.currentGraph.graph.dasharray = '';
KhanUtil.currentGraph.graph.update();
" />
Solid line
</label>
<label>
<input name="dashradio" type="radio" value="dashed" onClick="javascript:
KhanUtil.currentGraph.graph.dasharray = '- ';
KhanUtil.currentGraph.graph.update();
" />
Dashed line
</label>
</form>
<div class="graphie" id="grid">
graphInit({
range: 11,
scale: 20,
axisArrows: "<->",
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.line1 = bogusShape;
graph.shading = bogusShape;
graph.update = function() {
graph.line1.remove();
graph.shading.remove();
if ( abs( graph.pointB.coord[0] - graph.pointA.coord[0] ) > 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] < graph.pointB.coord[0] ) ? graph.shadetop : !graph.shadetop ) ? 11 : -11;
style({ stroke: BLUE, strokeWidth: 2, strokeDasharray: graph.dasharray }, function() {
graph.line1 = line( [ -11, -11 * slope + yint ], [ 11, 11 * slope + yint ] ).toBack();
});
style({ fill: BLUE, stroke: null, opacity: KhanUtil.FILL_OPACITY }, function() {
graph.shading = 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] < graph.pointA.coord[1] ) ? graph.shadetop : !graph.shadetop ) ? 11 : -11;
style({ stroke: BLUE, strokeWidth: 2, strokeDasharray: graph.dasharray }, function() {
graph.line1 = line( [ x, -11 ], [ x, 11 ] ).toBack();
});
style({ fill: BLUE, stroke: null, opacity: KhanUtil.FILL_OPACITY }, function() {
graph.shading = path([ [ x, -11 ], [ x, 11 ], [ shadeEdge, 11 ], [ shadeEdge, -11 ] ]);
});
}
};
// 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;
}
}
graph.dasharray = "";
graph.shadetop = true;
graph.update();
graph.pointA.toFront();
graph.pointB.toFront();
</div>
</div>
<div class="solution" data-type="multiple">
<div class="sol" data-type="custom">
<div class="instruction">
Drag the two points to move the line into the correct position.
</div>
<div class="guess">[ graph.pointA.coord, graph.pointB.coord,
graph.pointA.coord[0] > graph.pointB.coord[0] ? graph.shadetop : !graph.shadetop,
graph.dasharray === "- " ? false : true ]
</div>
<div class="validator-function">
var slope = ( guess[1][1] - guess[0][1] ) / ( guess[1][0] - guess[0][0] );
var yint = slope * ( 0 - guess[0][0] ) + guess[0][1];
return abs( SLOPE - slope ) < 0.001
&& abs( YINT - yint ) < 0.001
&& guess[2] === LESS_THAN
&& guess[3] === INCLUSIVE;
</div>
<div class="show-guess">
graph.pointA.setCoord( guess[0] );
graph.pointB.setCoord( guess[1] );
graph.shadetop = graph.pointA.coord[0] > graph.pointB.coord[0] ? guess[2] : !guess[2];
if ( guess[3] ) {
graph.dasharray = "";
$( "input[name=dashradio][value=solid]" ).attr( "checked", true );
} else {
graph.dasharray = "- ";
$( "input[name=dashradio][value=dashed]" ).attr( "checked", true );
}
graph.update();
</div>
</div>
<div class="sol" data-type="set">
<p class="input-format">
<label><span class="checkbox">POINT_1_SOLUTION</span><code>(<var>POINT_1[0]</var>, <var>POINT_1[1]</var>)</code> is a solution</label><br />
<label><span class="checkbox">POINT_2_SOLUTION</span><code>(<var>POINT_2[0]</var>, <var>POINT_2[1]</var>)</code> is a solution</label>
</p>
</div>
<div class="example">graph the inequality</div>
<div class="example">make sure the correct side is shaded</div>
<div class="example">make sure the line is correctly shown as solid or dashed</div>
</div>
</div>
</div>
<div class="hints">
<div data-if="STD_FORM" data-unwrap>
<p>
Convert <code><var>expr([ "+", [ "*", A, "x" ], [ "*", B, "y" ] ])</var> <var>STD_FORM_COMP</var> <var>C</var></code>
to slope-intercept form by solving for <code>y</code>.
</p>
<div>
<p>
<var>A < 0 ? "Add" : "Subtract"</var> <code><var>abs( A )</var>x</code> <var>A < 0 ? "to" : "from"</var> both sides:
</p><p>
<code>\qquad <var>expr( [ "*", B, "y" ] )</var> <var>STD_FORM_COMP</var> <var>expr([ "+", [ "*", -A, "x" ], C ])</var></code>
</p>
</div>
<div data-if="B !== 1">
<p>
Divide both sides by <code><var>B</var></code><span data-if="B < 0">. Since you're multiplying or dividing by a negative number, <strong>don't forget</strong> to flip the inequality sign</span>:
</p><p>
<code>\qquad y <var>COMP</var> <var>expr([ "+", "\\dfrac{" + expr([ "*", -A, "x" ]) + "}{" + B + "}", "\\dfrac{" + C + "}{" + B + "}" ])</var></code>
</p>
</div>
</div>
<div>
<p>
<code>\qquad y <var>COMP</var> \color{purple}{<var>PRETTY_SLOPE</var>} x \color{<var>GREEN</var>}{+ <var>YINT</var>}</code>
</p><p>
The y-intercept is <code class="hint_green"><var>YINT</var></code> and the slope is <code class="hint_purple"><var>decimalFraction( SLOPE, true, true )</var></code>.
Since the y-intercept is <code class="hint_green"><var>YINT</var></code>, the line must pass through the point <code class="hint_green">(0, <var>YINT</var>)</code>.
</p>
<div class="graphie" data-update="grid">
style({ stroke: GREEN, fill: GREEN }, function() {
graph.yint = circle( [ 0, YINT ], 0.2 ).toBack();
});
</div>
</div>
<div>
<p>
The slope is <code class="hint_purple"><var>decimalFraction( SLOPE, true, true )</var></code>. Remember that
the slope tells you rise over run. So in this case for every
<code><var>abs( SLOPE_FRAC[0] )</var></code>
<var>"position" + ( abs( SLOPE_FRAC[0] ) !== 1 ? "s" : "" )</var>
you move
<span data-if="SLOPE_FRAC[0] < 0"><em>down</em> (because it's negative)</span>
<span data-else>up</span>
you must also move
<code><var>SLOPE_FRAC[1]</var></code>
<var>"position" + ( abs( SLOPE_FRAC[1] ) !== 1 ? "s" : "" )</var>
to the right. So the line must also pass through <code class="hint_purple">(<var>SLOPE_FRAC[1]</var>, <var>YINT + SLOPE_FRAC[0]</var>)</code>
</p>
<div class="graphie" data-update="grid">
style({
stroke: "purple",
strokeWidth: 2,
arrows: "->"
}, function() {
path([ [ 0, YINT ], [ 0, YINT + SLOPE_FRAC[0] ] ]).toBack();
path([ [ 0, YINT + SLOPE_FRAC[0] ], [ SLOPE_FRAC[1], YINT + SLOPE_FRAC[0] ] ]).toBack();
});
label( [ 0, YINT + SLOPE_FRAC[0] / 2 ], abs( SLOPE_FRAC[0] ) + " \\text{ " + ( SLOPE_FRAC[0] < 0 ? "down" : "up" ) + "} \\quad", "left", { color: "purple" } );
label( [ 0, YINT + SLOPE_FRAC[0] ], SLOPE_FRAC[1] + " \\text{ right}", ( SLOPE_FRAC[0] < 0 ? "below right" : "above right" ), { color: "purple" } );
graph.yint.toBack();
</div>
</div>
<p>
Since our inequality has a <var>LESS_THAN ? "less-than" : "greater-than"</var><var>INCLUSIVE ? " or equal to" : ""</var> sign, that means that any point
<strong><var>LESS_THAN ? "below" : "above"</var></strong> the line is a solution to the inequality, so the area <var>LESS_THAN ? "below" : "above"</var>
the line should be shaded.
</p>
<div>
<p data-if="INCLUSIVE">
Note that since the sign is <var>LESS_THAN ? "less-than" : "greater-than"</var> or <strong>equal to</strong>, any point on the line is also a solution, so the
line should be solid.
</p>
<p data-else>
Note that since the sign is <var>LESS_THAN ? "less-than" : "greater-than"</var> (and not equal to), any point on the line is <strong>not</strong> part of the solution,
so the line should be dashed to indicate this.
</p>
</div>
<div>
<p>
Plot each of the points to determine which are solutions:
<code class="hint_pink">(<var>POINT_1[0]</var>, <var>POINT_1[1]</var>)</code> and
<code class="hint_pink">(<var>POINT_2[0]</var>, <var>POINT_2[1]</var>)</code>
</p>
<div class="graphie" data-update="grid">
style({ stroke: PINK, fill: PINK }, function() {
circle( [ POINT_1[0], POINT_1[1] ], 0.2 ).toBack();
circle( [ POINT_2[0], POINT_2[1] ], 0.2 ).toBack();
});
</div>
</div>
<div>
<p data-if="POINT_1_SOLUTION && !POINT_2_SOLUTION">
<code class="hint_pink">(<var>POINT_1[0]</var>, <var>POINT_1[1]</var>)</code> is a solution because it's inside the shaded area, but
<code class="hint_pink">(<var>POINT_2[0]</var>, <var>POINT_2[1]</var>)</code> is not.
</p>
<p data-else-if="POINT_2_SOLUTION && !POINT_1_SOLUTION">
<code class="hint_pink">(<var>POINT_1[0]</var>, <var>POINT_1[1]</var>)</code> is outside the shaded area, so it's not a solution, but
<code class="hint_pink">(<var>POINT_2[0]</var>, <var>POINT_2[1]</var>)</code> is a solution.
</p>
<p data-else-if="POINT_1_SOLUTION && POINT_2_SOLUTION"> Both
<code class="hint_pink">(<var>POINT_1[0]</var>, <var>POINT_1[1]</var>)</code> and
<code class="hint_pink">(<var>POINT_2[0]</var>, <var>POINT_2[1]</var>)</code> are solutions because they're inside the shaded area.
</p>
<p data-else> Neither
<code class="hint_pink">(<var>POINT_1[0]</var>, <var>POINT_1[1]</var>)</code> nor
<code class="hint_pink">(<var>POINT_2[0]</var>, <var>POINT_2[1]</var>)</code> are solutions because they are not within the shaded area.
</p>
</div>
</div>
</div>
</body>
</html>