< 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.
投影片
Sign up free to view the lesson slides
Step through every slide for this topic — plus flashcards and revision notes — with a free account.
