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Inequalities

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  Edexcel IAL Further Pure Mathematics 2 (WFM02)

Inequalities

The big idea: You may only multiply an inequality by something you know is positive. Every method in this chapter is a way of respecting that one rule when the expression contains x.

Summary — What This Chapter Covers
  • Inequalities with x in a denominator: never multiply by an expression whose sign you don't know.
  • Method 1: collect everything on one side as a single fraction, find the critical values, and use a sign table.
  • Method 2: multiply both sides by the square of the denominator (always positive), then solve the polynomial inequality.
  • Graphs show at a glance where one side is above the other — use them to check, or to solve after finding intersections.
  • Modulus inequalities: split into cases, or square both sides when both sides are non-negative.
1. Algebraic Inequalities
Critical values

Why you can't just "cross-multiply"

If you multiply both sides of an inequality by a negative number, the inequality sign flips. With an expression like (x + 1) you don't know whether it is positive or negative, so multiplying by it is not allowed.

The safe method: move everything to one side, write it as a single fraction, and factorise the numerator and denominator. The critical values are where the numerator is zero (the expression is 0) and where the denominator is zero (the expression is undefined). Between consecutive critical values the sign cannot change, so test one value in each interval.

Method f(x) > g(x)  ⇔  f(x) − g(x) > 0  → single fraction N(x)/D(x)  → critical values from N(x) = 0 and D(x) = 0  → sign table
Alternative Multiply both sides by D(x)2 (never negative), giving a polynomial inequality; then remember D(x) ≠ 0.
Worked Example

Using algebra, solve 3/(x + 1) < x − 1.

Practice Question 1

Solve (x + 3)/(x − 1) ≤ 2.

2. Using Graphs to Solve Inequalities
Graphs

Reading the answer off a sketch

f(x) > g(x) wherever the graph of y = f(x) lies above the graph of y = g(x). Sketch both, find the x-coordinates of the points of intersection algebraically, and read off the intervals. Vertical asymptotes of either graph are also boundaries of the intervals.

Worked Example

Solve |2x − 3| > x.

Exam habit
A quick sketch is the fastest way to catch a lost interval. If your algebra gives one interval but the sketch shows two regions where one graph is above the other, recheck your critical values.
3. Modulus Inequalities
Modulus

Cases, or squaring

both sides are non-negative

Worked Example

Solve |x2 − 1| > 2(x + 1).

Practice Question 2

Solve |x − 1| > |2x + 1|.

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More Further Pure Mathematics 2 topics

Algebraic inequalities
Maclaurin series
The method of differences
Separable variables and families of solution curves
Exponential form and Euler's relation
Polar coordinates and equations
Loci in the Argand diagram
Using graphs to solve inequalities
Partial fractions and the method of differences
De Moivre's theorem