< less than, > greater than, \leq less than or equal to, \geq greater than or equal to.
-2 < x \leq 3 means x is greater than -2 and at most 3. Integers satisfying it: -1, 0, 1, 2, 3.
Solving linear inequalities
Solve like an equation, doing the same to both sides. The answer is a range: 3x + 2 > 11 gives x > 3.
Reverse the inequality sign if you multiply or divide by a negative number: -2x < 8 gives x > -4.
Number lines and set notation
A filled circle shows the end value is included (\leq, \geq); an open circle shows it is excluded (<, >).
Write a double inequality such as -1 \leq x < 4 as one range on a number line.
Quadratic inequalities
Solve x2 - x - 6 < 0: factorise (x - 3)(x + 2) < 0. Roots -2 and 3. The parabola is below the axis between the roots, so -2 < x < 3.
For x2 > 9: x < -3 or x > 3.
Where the quadratic is negative
Graphical inequalities
Draw a solid boundary for ≤ or ≥, and a dashed boundary for < or >. Test a point not on the boundary to choose the required side; the origin works only if it is off the boundary.
Regions defined by several inequalities: shade out the unwanted regions and leave the required region unshaded, or as the question states.
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