randRangeNonZero( -10, 10 ) randRangeNonZero( -10, 10 ) sortNumbers([-1*A, -1*B]) ROOTS[0] ROOTS[1] A * B "x" "x^2" parens(plus(X, A)) parens(plus(X, B))

For what values of x does

plus(XX, (A+B)+X) < -C

R1  < x <  R2

First, move the term over to the other side of the inequality
plus(XX, (A+B)+X, C) < 0

Now, let's factor the quadratic equation
mult(LEFT, RIGHT) < 0

either  LEFT < 0 and RIGHT > 0 or
LEFT > 0 and RIGHT < 0

either  X < -A and X > -B or
X > -A and X < -B

which we can graph, like so:

init({ range: [ [0, 24], [-2, 1.25] ], scale: [20,20], }); colors = shuffle(["#1F78B4", "#B2DF8A", "#33A02C","#A6CEE3"], 2) style({ stroke: "#666" }, function() { numberLine( -12, 12, 2 ); }); style({ stroke: colors[0], strokeWidth:5 }, function() { line([12+R1,1], [12+R2,1]); label([12+(R1+R2)/2,.5], R1 + " < x < "+R2, "above"); }); style({ stroke: colors[1], strokeWidth: 5 }, function() { line( [0, 1], [12+R1, 1] ); line( [12+R2, 1], [24, 1] ); });

By plugging in a value for x we can see which of the inequalities fits