randRangeNonZero( -5, 5 ) randRange( -5, 5 )
randRangeNonZero( -5, 5 ) randRangeNonZero( -5, 5 )
(Y - YINT_1) / X (Y - YINT_2) / X toFraction( SLOPE_1, 0.001 ) toFraction( SLOPE_2, 0.001 )
SLOPE_1 === 1 ? "" : ( SLOPE_1 === -1 ? "-" : fraction( SLOPE_FRAC_1, SLOPE_FRAC_1, true, true ) ) SLOPE_2 === 1 ? "" : ( SLOPE_2 === -1 ? "-" : fraction( SLOPE_FRAC_2, SLOPE_FRAC_2, true, true ) ) randRangeNonZero( -3, 3 ) randRangeNonZero( -3, 3 ) SLOPE_FRAC_1 * -MULT_1 SLOPE_FRAC_2 * -MULT_2 SLOPE_FRAC_1 * MULT_1 SLOPE_FRAC_2 * MULT_2 SLOPE_FRAC_1 * YINT_1 * MULT_1 SLOPE_FRAC_2 * YINT_2 * MULT_2 randFromArray([ true, false ]) randFromArray([ true, false ]) randFromArray([ "<", ">", "≤", "≥" ]) randFromArray([ "<", ">", "≤", "≥" ]) B_1 < 0 ? { "<": ">", ">": "<", "≤": "≥", "≥": "≤" }[ COMP_1 ] : COMP_1 B_2 < 0 ? { "<": ">", ">": "<", "≤": "≥", "≥": "≤" }[ COMP_2 ] : COMP_2 COMP_1 == "<" || COMP_1 == "≤" COMP_2 == "<" || COMP_2 == "≤" COMP_1 == "≥" || COMP_1 == "≤" COMP_2 == "≥" || COMP_2 == "≤"

Graph the following system of inequalities:

expr([ "+", [ "*", A_1, "x" ], [ "*", B_1, "y" ] ]) STD_FORM_COMP_1 C_1

y COMP_1 PRETTY_SLOPE_1 x + YINT_1

expr([ "+", [ "*", A_2, "x" ], [ "*", B_2, "y" ] ]) STD_FORM_COMP_2 C_2

y COMP_2 PRETTY_SLOPE_2 x + YINT_2

Inequality 1:
Inequality 2:
Drag the points to move both lines into the correct positions.
[ graph.pointA.coord, graph.pointB.coord, graph.pointA.coord > graph.pointB.coord ? graph.shadetop1 : !graph.shadetop1, graph.dasharray1 === "- " ? false : true, graph.pointC.coord, graph.pointD.coord, graph.pointC.coord > graph.pointD.coord ? graph.shadetop2 : !graph.shadetop2, graph.dasharray2 === "- " ? false : true ]
var slope1 = ( guess - guess ) / ( guess - guess ); var yint1 = slope1 * ( 0 - guess ) + guess; var slope2 = ( guess - guess ) / ( guess - guess ); var yint2 = slope2 * ( 0 - guess ) + guess; return abs( SLOPE_1 - slope1 ) < 0.001 && abs( YINT_1 - yint1 ) < 0.001 && guess === LESS_THAN_1 && guess === INCLUSIVE_1 && abs( SLOPE_2 - slope2 ) < 0.001 && abs( YINT_2 - yint2 ) < 0.001 && guess === LESS_THAN_2 && guess === INCLUSIVE_2;
graph.pointA.setCoord( guess ); graph.pointB.setCoord( guess ); graph.pointC.setCoord( guess ); graph.pointD.setCoord( guess ); graph.shadetop1 = graph.pointA.coord > graph.pointB.coord ? guess : !guess; graph.shadetop2 = graph.pointC.coord > graph.pointD.coord ? guess : !guess; if ( guess ) { graph.dasharray1 = ""; jQuery( "input[name=dashradio1][value=solid]" ).attr( "checked", true ); } else { graph.dasharray1 = "- "; jQuery( "input[name=dashradio1][value=dashed]" ).attr( "checked", true ); } if ( guess ) { graph.dasharray2 = ""; jQuery( "input[name=dashradio2][value=solid]" ).attr( "checked", true ); } else { graph.dasharray2 = "- "; jQuery( "input[name=dashradio2][value=dashed]" ).attr( "checked", true ); } graph.update();
graph the inequalities
make sure the correct sides are shaded
make sure each line is correctly shown as solid or dashed

Convert the first inequality, expr([ "+", [ "*", A_1, "x" ], [ "*", B_1, "y" ] ]) STD_FORM_COMP_1 C_1, to slope-intercept form by solving for y.

A_1 < 0 ? "Add" : "Subtract" abs( A_1 )x A_1 < 0 ? "to" : "from" both sides:

\qquad expr( [ "*", B_1, "y" ] ) STD_FORM_COMP_1 expr([ "+", [ "*", -A_1, "x" ], C_1 ])

Divide both sides by B_1. Since you're multiplying or dividing by a negative number, don't forget to flip the inequality sign:

\qquad y COMP_1 expr([ "+", "\\dfrac{" + expr([ "*", -A_1, "x" ]) + "}{" + B_1 + "}", "\\dfrac{" + C_1 + "}{" + B_1 + "}" ])

\qquad y COMP_1 \color{purple}{PRETTY_SLOPE_1} x \color{gray}{+ YINT_1}

The y-intercept is YINT_1 and the slope is decimalFraction( SLOPE_1, true, true ). Since the y-intercept is YINT_1, the line must pass through the point (0, YINT_1).

The slope is decimalFraction( SLOPE_1, true, true ). Remember that the slope tells you rise over run. So in this case for every abs( SLOPE_FRAC_1 ) "position" + ( abs( SLOPE_FRAC_1 ) !== 1 ? "s" : "" ) you move down (because it's negative) up you must also move SLOPE_FRAC_1 "position" + ( abs( SLOPE_FRAC_1 ) !== 1 ? "s" : "" ) to the right. So the line must also pass through (SLOPE_FRAC_1, YINT_1 + SLOPE_FRAC_1)

Since our inequality has a LESS_THAN_1 ? "less-than" : "greater-than"INCLUSIVE_1 ? " or equal to" : "" sign, that means that any point LESS_THAN_1 ? "below" : "above" the line is a solution to the inequality, so the area LESS_THAN_1 ? "below" : "above" the line should be shaded.

Note that since the sign is LESS_THAN_1 ? "less-than" : "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_1 ? "less-than" : "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.

Convert the second inequality, expr([ "+", [ "*", A_2, "x" ], [ "*", B_2, "y" ] ]) STD_FORM_COMP_2 C_2, to slope-intercept form by solving for y.

A_2 < 0 ? "Add" : "Subtract" abs( A_2 )x A_2 < 0 ? "to" : "from" both sides:

\qquad expr( [ "*", B_2, "y" ] ) STD_FORM_COMP_2 expr([ "+", [ "*", -A_2, "x" ], C_2 ])

Divide both sides by B_2. Since you're multiplying or dividing by a negative number, don't forget to flip the inequality sign:

\qquad y COMP_2 expr([ "+", "\\dfrac{" + expr([ "*", -A_2, "x" ]) + "}{" + B_2 + "}", "\\dfrac{" + C_2 + "}{" + B_2 + "}" ])

\qquad y COMP_2 \color{purple}{PRETTY_SLOPE_2} x \color{gray}{+ YINT_2}

The y-intercept is YINT_2 and the slope is decimalFraction( SLOPE_2, true, true ). Since the y-intercept is YINT_2, the line must pass through the point (0, YINT_2).

The slope is decimalFraction( SLOPE_2, true, true ). Remember that the slope tells you rise over run. So in this case for every abs( SLOPE_FRAC_2 ) "position" + ( abs( SLOPE_FRAC_2 ) !== 1 ? "s" : "" ) you move down (because it's negative) up you must also move SLOPE_FRAC_2 "position" + ( abs( SLOPE_FRAC_2 ) !== 1 ? "s" : "" ) to the right. So the line must also pass through (SLOPE_FRAC_2, YINT_2 + SLOPE_FRAC_2)

Since our inequality has a LESS_THAN_2 ? "less-than" : "greater-than"INCLUSIVE_2 ? " or equal to" : "" sign, that means that any point LESS_THAN_2 ? "below" : "above" the line is a solution to the inequality, so the area LESS_THAN_2 ? "below" : "above" the line should be shaded.

Note that since the sign is LESS_THAN_2 ? "less-than" : "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_2 ? "less-than" : "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.