tabulate( function() { return randRange( 2, 9 ) * ( rand( 3 ) > 0 ? 1 : -1 ); } , 2 ) tabulate( function(i) { return fraction( 1, COEF[i] ); }, 2 ) randFromArray( [ "-", "+" ], 2) tabulate( function() { return rand( 3 ) > 0 ? randRange( 2, 9 ) : 0; }, 2 ) tabulate( function(i) { return LEFT_INT[i] * ( SIGN[i] === "+" ? -1 : 1 ); }, 2 ) randFromArray( [ "<", ">", "≤", "≥" ], 2 ) tabulate( function(i) { return randRange( 1, 6 ) * abs( COEF[i] ) + ( SIGN[i] === "+" ? 1 : -1 ) * LEFT_INT[i]; }, 2 ) randVar() tabulate( function(i) { return getComp( COEF[i], COMP[i] ); }, 2 ) tabulate( function(i) { return fraction( RIGHT_INT[i] + ADD_TO_SIMPLIFY[i], COEF[i] ); }, 2 ) tabulate( function(i) { return ( RIGHT_INT[i] + ADD_TO_SIMPLIFY[i] ) / COEF[i]; }, 2 ) tabulate( function(i) { return COMP_SOLUTION[i] === "≤" || COMP_SOLUTION[i] === "≥"; }, 2 ) tabulate( function(i) { return COMP_SOLUTION[i] === "≤" || COMP_SOLUTION[i] === "<"; }, 2 ) [BLUE, PINK] randRange(0, 1) === 0 [ $._("or"), $._("and") ][IS_OR ? 0 : 1] IS_OR && ((LESS_THAN[0] && !LESS_THAN[1] && SOLUTION[0] >= SOLUTION [1]) || (!LESS_THAN[0] && LESS_THAN[1] && SOLUTION[0] <= SOLUTION [1])) !IS_OR && ((LESS_THAN[0] && !LESS_THAN[1] && SOLUTION[1] > SOLUTION[0]) || (!LESS_THAN[0] && LESS_THAN[1] && SOLUTION[0] > SOLUTION[1]) || (LESS_THAN[0] !== LESS_THAN[1] && SOLUTION[0] === SOLUTION[1] && (!INCLUSIVE[0] || !INCLUSIVE[1]))) !IS_OR && SOLUTION[0] === SOLUTION[1] && COMP.indexOf("≤") !== -1 && COMP.indexOf("≥") !== -1 tabulate( function() { return randRange( 2, 9 ) * ( rand( 3 ) > 0 ? 1 : -1 ); }, 2 ) [ round((RIGHT_INT[0] - ADD_TO_SIMPLIFY[0]) / COEF[0]), round((RIGHT_INT[1] - ADD_TO_SIMPLIFY[1]) / COEF[1]) ] (function() { if ( LESS_THAN[0] && LESS_THAN[1] ) { if ( SOLUTION[0] === SOLUTION[1] ) { return INCLUSIVE[0] ? 1 : 2; } return SOLUTION[0] > SOLUTION[1] ? 1 : 2; } else if ( !LESS_THAN[0] && !LESS_THAN[1] ) { if ( SOLUTION[0] === SOLUTION[1] ) { return INCLUSIVE[0] ? 1 : 2; } return SOLUTION[0] < SOLUTION[1] ? 1 : 2; } return 0; })() CONTAINS === 1 ? 2 : 1

Solve for VARIABLE_NAME:

\blue{COEF[0] + VARIABLE_NAMESIGN[0] + LEFT_INT[0]COMP[0] + RIGHT_INT[0]} OR \pink{COEF[1] + VARIABLE_NAMESIGN[1] + LEFT_INT[1]COMP[1] + RIGHT_INT[1]}

VARIABLE_NAME = SOLUTION[0] No solution. All real numbers. VARIABLE_NAME + COMP_SOLUTION[CONTAINS - 1] + SOLUTION[CONTAINS - 1] VARIABLE_NAME + COMP_SOLUTION[IS_CONTAINED - 1] + SOLUTION[IS_CONTAINED - 1] VARIABLE_NAME + COMP_SOLUTION[0] + SOLUTION[0] OR VARIABLE_NAME + COMP_SOLUTION[1] + SOLUTION[1]

  • VARIABLE_NAME + COMP_SOLUTION[0] + FAKE_ANSWER[0] OR VARIABLE_NAME + COMP_SOLUTION[1] + FAKE_ANSWER[1]
  • VARIABLE_NAME + COMP_SOLUTION[0] + SOLUTION[0] OR VARIABLE_NAME + COMP_SOLUTION[1] + randRangeExclude(-9, 9, [0, SOLUTION[0]])
  • VARIABLE_NAME + COMP_SOLUTION[0] + randRangeExclude(-9, 9, [0, SOLUTION[1]]) OR VARIABLE_NAME + COMP_SOLUTION[1] + SOLUTION[1]
  • VARIABLE_NAME + COMP_SOLUTION[0] + FAKE_ANSWER[0]
  • VARIABLE_NAME + COMP_SOLUTION[0] + SOLUTION[0] OR VARIABLE_NAME + COMP_SOLUTION[1] + SOLUTION[1]
  • VARIABLE_NAME + COMP[0] + SOLUTION[0] OR VARIABLE_NAME + COMP_SOLUTION[1] + SOLUTION[1]
  • VARIABLE_NAME + COMP[0] + FAKE_ANSWER[0] OR VARIABLE_NAME + COMP_SOLUTION[1] + FAKE_ANSWER[1]
  • VARIABLE_NAME + COMP_SOLUTION[0] + SOLUTION[0] OR VARIABLE_NAME + COMP[1] + SOLUTION[1]
  • VARIABLE_NAME + COMP_SOLUTION[0] + FAKE_ANSWER[0] OR VARIABLE_NAME + COMP[1] + FAKE_ANSWER[1]
  • VARIABLE_NAME + COMP[0] + SOLUTION[0] OR VARIABLE_NAME + COMP[1] + SOLUTION[1]
  • VARIABLE_NAME + COMP[0] + FAKE_ANSWER[0] OR VARIABLE_NAME + COMP[1] + FAKE_ANSWER[1]
  • VARIABLE_NAME + COMP_SOLUTION[0] + WRONG_SIMPLIFICATION[0] OR VARIABLE_NAME + COMP_SOLUTION[1] + WRONG_SIMPLIFICATION[1]
  • All real numbers.
  • No solution.
  • VARIABLE_NAME = SOLUTION[0]

The first inequality can be simplified to: \blue{VARIABLE_NAME + COMP_SOLUTION[0] + SOLUTION[0]}.

The second inequality can be simplified to: \pink{VARIABLE_NAME + COMP_SOLUTION[1] + SOLUTION[1]}.

The two inequalities are represented on the number line below:

var start = min( SOLUTION[0], SOLUTION[1] ) - randRange( 2, 5 ); var end = max( SOLUTION[0], SOLUTION[1] ) + randRange( 2, 5 ); init({ range: [ [ start - 1, end + 1 ], [ -1, 1 ] ], scale: 28 }); numberLine( start, end, null, start ); // Draw both inequalities var y_placement = [ 0.05, -0.05 ]; for ( var i = 0; i < 2; i++ ) { style({ stroke: COLOR[i], fill: COLOR[i], strokeWidth: 3.5, arrows: "->" }); path([ [ SOLUTION[i] + 0.15 * ( LESS_THAN[i] ? -1 : 1 ), y_placement[i] ], [ LESS_THAN[i] ? start : end, y_placement[i] ] ]); style({ stroke: COLOR[i], fill: INCLUSIVE[i] ? COLOR[i] : null }); circle( [ SOLUTION[i], y_placement[i] ], 0.2 ); }

Since this is an "or" inequality, the solution is the part of the number line which is covered by either of the graphs of the inequalities.

The combined graphs of the inequalities span the entire number line, therefore the solution is "All real numbers."

Notice how the first inequality is completely included by the second inequality. Therefore the answer is:

Notice how the second inequality is completely included by the first inequality. Therefore the answer is:

\color{COLOR[CONTAINS - 1]}{VARIABLE_NAME + COMP_SOLUTION[CONTAINS - 1] + SOLUTION[CONTAINS - 1]}

Therefore, since the graphs of the inequalities do not intersect, the solution is:

\blue{VARIABLE_NAME + COMP_SOLUTION[0] + SOLUTION[0]} or \pink{VARIABLE_NAME + COMP_SOLUTION[1] + SOLUTION[1]}

The solution to an inequality with the word "and" is the intersection of the graphs of the inequalities.

Therefore, the solution is:

\purple{VARIABLE_NAME = SOLUTION[0]}

Since the graphs of the inequalities do not intersect, there is no solution.

Since the second inequality is completely included by the first inequality, their intersection is the second inequality. Therefore the answer is:

Since the first inequality is completely included by the second inequality, their intersection is the first inequality. Therefore the answer is:

\color{COLOR[IS_CONTAINED - 1]}{VARIABLE_NAME + COMP_SOLUTION[IS_CONTAINED - 1] + SOLUTION[IS_CONTAINED - 1]}

Therefore, the solution is:

\blue{VARIABLE_NAME + COMP_SOLUTION[0] + SOLUTION[0]} and \pink{VARIABLE_NAME + COMP_SOLUTION[1] + SOLUTION[1]}