reduce( randRangeNonZero( -5, 5 ), randRange( 1, 5 ) ) randRangeNonZero( max( -10, -10 - SLOPE_FRAC[0] ), min( 10, 10 - SLOPE_FRAC[0] ) ) SLOPE_FRAC[0] / SLOPE_FRAC[1] fractionVariable(SLOPE_FRAC[0], SLOPE_FRAC[1], "x") randRangeNonZero( -3, 3 ) SLOPE_FRAC[0] * -MULT SLOPE_FRAC[1] * MULT SLOPE_FRAC[1] * YINT * MULT randFromArray([ true, false ]) randFromArray([ "<", ">", "≤", "≥" ]) B < 0 ? { "<": ">", ">": "<", "≤": "≥", "≥": "≤" }[ COMP ] : COMP COMP === "<" || COMP === "≤" COMP === "≥" || COMP === "≤"
(function() { // Create a point on the line var minN = floor(9 / abs(SLOPE_FRAC[1])); var n = randRange(-minN, minN); return [n * SLOPE_FRAC[1], YINT+ n * SLOPE_FRAC[0]]; })() shuffle([ [randRangeExclude(-9, 9, [-3, -2, -1, 0]), randRangeExclude(-9, 9, [-1, -2])], [randRangeExclude(-9, 9, [-3, -2, -1, 0]), randRangeExclude(-9, 9, [-1, -2])], rand(2) <1 ? EDGE_POINT : [randRangeExclude(-9, 9, [-3, -2, -1, 0]), randRangeExclude(-9, 9, [-1, -2])] ]).slice(1) POINT_1[1] === SLOPE * POINT_1[0] + YINT POINT_2[1] === SLOPE * POINT_2[0] + YINT (function() { if (POINT_1_ON_LINE) { return INCLUSIVE; } return (POINT_1[1] < SLOPE * POINT_1[0] + YINT) ? LESS_THAN : !LESS_THAN; })() (function() { if (POINT_2_ON_LINE) { return INCLUSIVE; } return (POINT_2[1] < SLOPE * POINT_2[0] + YINT) ? LESS_THAN : !LESS_THAN; })()

Graph the following inequality and determine if the points `(POINT_1[0], POINT_1[1])` or `(POINT_2[0], POINT_2[1])` are solutions:

`expr([ "+", [ "*", A, "x" ], [ "*", B, "y" ] ]) STD_FORM_COMP C`

`y COMP PRETTY_SLOPE + YINT`

- Solid line
- Dashed line

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 ] ]); }); } graph.pointA.toFront(); graph.pointB.toFront(); }; graph.showCorrect = function() { graph.pointA.setCoord([0, YINT]); graph.pointB.setCoord([SLOPE_FRAC[1], YINT + SLOPE_FRAC[0]]); graph.shadetop = graph.pointA.coord[0] > graph.pointB.coord[0] ? LESS_THAN : !LESS_THAN; if (INCLUSIVE) { graph.dasharray = ''; $('input[name=dashradio][value=solid]').attr('checked', true); } else { graph.dasharray = '- '; $('input[name=dashradio][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; } } graph.dasharray = ""; graph.shadetop = true; graph.update(); graph.pointA.toFront(); graph.pointB.toFront();
Drag the two points to move the line into the correct position.
[ graph.pointA.coord, graph.pointB.coord, graph.pointA.coord[0] > graph.pointB.coord[0] ? graph.shadetop : !graph.shadetop, graph.dasharray === "- " ? false : true ]
if (_.isEqual(guess, [[-5, 5], [5, 5], false, true])) { return ""; } 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;
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();

- POINT_1_SOLUTION `(POINT_1[0], POINT_1[1])` is a solution
- POINT_2_SOLUTION `(POINT_2[0], POINT_2[1])` is a solution

Convert `expr([ "+", [ "*", A, "x" ], [ "*", B, "y" ] ]) STD_FORM_COMP C` to slope-intercept form by solving for `y`.

Add `abs( A )x` to both sides:

Subtract `abs( A )x` from both sides:

`\qquad expr( [ "*", B, "y" ] ) STD_FORM_COMP expr([ "+", [ "*", -A, "x" ], C ])`

Divide both sides by `B`. Since you're multiplying or dividing by a negative number, don't forget to flip the inequality sign:

`\qquad y COMP expr([ "+", "\\dfrac{" + expr([ "*", -A, "x" ]) + "}{" + B + "}", "\\dfrac{" + C + "}{" + B + "}" ])`

`\qquad y COMP \purple{PRETTY_SLOPE} \green{+ YINT}`

The y-intercept is `YINT` and the slope is `decimalFraction( SLOPE, true, true )`. Since the y-intercept is `YINT`, the line must pass through the point `(0, YINT)`.

style({ stroke: GREEN, fill: GREEN }, function() { graph.yint = circle( [ 0, YINT ], 0.2 ).toBack(); });

The slope is `decimalFraction( SLOPE, true, true )`. Remember that the slope tells you rise over run. For every step we take we must move `abs( SLOPE_FRAC[0] )` positions down (because it's negative)up. For every step we take we must move `abs( SLOPE_FRAC[0] )` position down (because it's negative)up. For every step we take we must also move `SLOPE_FRAC[1]` positions to the right. For every step we take we must also move `SLOPE_FRAC[1]` position to the right. So the line must also pass through `(SLOPE_FRAC[1], YINT + SLOPE_FRAC[0])`.

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();

Since our inequality has a less-than or equal to sign, that means that any point below the line is a solution to the inequality, so the area below the line should be shaded.

Since our inequality has a greater-than or equal to sign, that means that any point above the line is a solution to the inequality, so the area above the line should be shaded.

Note that since the sign is less-than or equal to, any point on the line is also a solution, so the line should be solid. Note that since the sign is greater-than or equal to, any point on the line is also a solution, so the line should be solid.

Note that since the sign is less-than (and not equal to), any point on the line is not part of the solution, so the line should be dashed to indicate this. Note that since the sign is greater-than (and not equal to), any point on the line is not part of the solution, so the line should be dashed to indicate this.

Plot each of the points to determine which are solutions: `\pink{(POINT_1[0], POINT_1[1])}` and `\pink{(POINT_2[0], POINT_2[1])}`.

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(); });

`\pink{(POINT_1[0], POINT_1[1])}` is on the line so it satisfies `y = PRETTY_SLOPE + YINT`.

Since `COMP` is inclusive, `\pink{(POINT_1[0], POINT_1[1])}` is a solution.

Since `COMP` is not inclusive, `\pink{(POINT_1[0], POINT_1[1])}` is not a solution.

`\pink{(POINT_1[0], POINT_1[1])}` is a solution because it's inside the shaded area.

`\pink{(POINT_1[0], POINT_1[1])}` is not a solution because it's outside the shaded area.

`\pink{(POINT_2[0], POINT_2[1])}` is on the line so it satisfies `y = PRETTY_SLOPE + YINT`.

Since `COMP` is inclusive, `\pink{(POINT_2[0], POINT_2[1])}` is a solution.

Since `COMP` is not inclusive, `\pink{(POINT_2[0], POINT_2[1])}` is not a solution.

`\pink{(POINT_2[0], POINT_2[1])}` is a solution because it's inside the shaded area.

`\pink{(POINT_2[0], POINT_2[1])}` is not a solution because it's outside the shaded area.