Quadratic and graphical inequalities

To solve a quadratic inequality, find where the curve crosses the x-axis, then use a sketch.

1. Solve the matching equation (= 0) to find the roots.
2. Sketch the curve: a U shape if x² is positive.
3. "< 0" is where the curve is below the x-axis; "> 0" is where it is above.

Example: solve x² − x − 6 < 0.

1. (x − 3)(x + 2) = 0, so the roots are x = −2 and x = 3
2. The U-shaped curve is below the x-axis between the roots
3. −2 < x < 3

When the curve is above the x-axis, the answer is in two parts, one on each side. Write it as two inequalities joined by "or", or in set notation.

Example: solve x² − x − 6 ≥ 0.

1. The roots are −2 and 3, as before
2. The curve is above (or on) the x-axis outside the roots
3. x ≤ −2 or x ≥ 3
4. In set notation: {x : x ≤ −2} ∪ {x : x ≥ 3}

Example: solve x² > 16.

1. x² − 16 > 0, and the roots are −4 and 4
2. Outside the roots: x < −4 or x > 4

{x : …} means "the set of x such that …", and ∪ means "or".

Each inequality in x and y is a region on one side of a line.

1. Draw each boundary line: solid for ≤ or ≥ (points on it count), dashed for < or > (they don't).
2. Test a point, such as (0, 0), to see which side satisfies each inequality.
3. Shade the region that satisfies them all (or what's unwanted, if the question says so).

Example: describe the region where y ≤ x + 2, y > 1 and x < 3, and list the points in it with whole-number coordinates.

1. Lines: y = x + 2 (solid), y = 1 (dashed), x = 3 (dashed)
2. It is a triangle with corners (−1, 1), (3, 1) and (3, 5)
3. Points: (0, 2), (1, 2), (2, 2), (1, 3), (2, 3), (2, 4)

Points on a dashed line, like (2, 1), don't count.