Solving Inequalities Graphically

How can I solve inequalities graphically?

Consider the inequality f(x) ≤ g(x), where f(x) and g(x) are functions of x

if we move g(x) to the LHS we get

f(x) – g(x) ≤ 0

Solve f(x) – g(x) = 0 to find the zeros of f(x) – g(x)

These correspond to the x-coordinates of the points of intersection of the graphs y = f(x) and y = g(x)

To solve the inequality we can use a graph

Graph y = f(x) – g(x) and label its zeros
Hence find the intervals of x that satisfy the inequality f(x) – g(x) ≤ 0

These are the intervals which satisfies the original inequality f(x) ≤ g(x)

This method is particularly useful when finding the intersections between the functions is difficult due to needing large x and y windows on your GDC

Be careful when rearranging inequalities!

Remember to flip the sign of the inequality when you multiply or divide both sides by a negative number
- e. 1 < 2 → [times both sides by (–1)] → –1 > –2 (sign flips)
Never multiply or divide by a variable as this could be positive or negative
- You can only multiply by a term if you are certain it is always positive (or always negative)
  - Such as $x^{2}, |x|, e^{x}$
Some functions reverse the inequality
- Taking reciprocals of positive values
  - $0 < x < y \Rightarrow \frac{1}{x} > \frac{1}{y}$
- Taking logarithms when the base is 0 < a < 1
  - $0 < x < y \Rightarrow \log_{a} (x) > \log_{a} (y)$
The safest way to rearrange is simply to add & subtract to move all the terms onto one side

Worked example

Use a GDC to solve the inequality $2 x^{3} < x^{5} - 2 x$ .

2-8-1-ib-aa-hl-graphical-inequalities-we-solution

Solving Inequalities Graphically (DP IB Maths: AA HL)

Revision Note

How can I solve inequalities graphically?

Be careful when rearranging inequalities!

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1. Number & Algebra

1.1 Number & Algebra Toolkit

1.1.1 Standard Form

1.1.2 Laws of Indices

1.1.3 Partial Fractions

1.2 Exponentials & Logs

1.2.1 Introduction to Logarithms

1.2.2 Laws of Logarithms

1.2.3 Solving Exponential Equations

1.3 Sequences & Series

1.3.1 Language of Sequences & Series

1.3.2 Arithmetic Sequences & Series

1.3.3 Geometric Sequences & Series

1.3.4 Applications of Sequences & Series

1.3.5 Compound Interest & Depreciation

1.4 Simple Proof & Reasoning

1.4.1 Proof

1.5 Further Proof & Reasoning

1.5.1 Proof by Induction

1.5.2 Proof by Contradiction

1.6 Binomial Theorem

1.6.1 Binomial Theorem

1.6.2 Extension of The Binomial Theorem

1.7 Permutations & Combinations

1.7.1 Counting Principles

1.7.2 Permutations & Combinations

1.8 Complex Numbers

1.8.1 Intro to Complex Numbers

1.8.2 Modulus & Argument

1.8.3 Introduction to Argand Diagrams

1.9 Further Complex Numbers

1.9.1 Geometry of Complex Numbers

1.9.2 Forms of Complex Numbers

1.9.3 Complex Roots of Polynomials

1.9.4 De Moivre's Theorem

1.9.5 Roots of Complex Numbers

1.10 Systems of Linear Equations

1.10.1 Systems of Linear Equations

1.10.2 Algebraic Solutions

2. Functions

2.1 Linear Functions & Graphs

2.1.1 Equations of a Straight Line

2.2 Quadratic Functions & Graphs

2.2.1 Quadratic Functions

2.2.2 Factorising & Completing the Square

2.2.3 Solving Quadratics

2.2.4 Quadratic Inequalities

2.2.5 Discriminants

2.3 Functions Toolkit

2.3.1 Language of Functions

2.3.2 Composite & Inverse Functions

2.3.3 Symmetry of Functions

2.3.4 Graphing Functions

2.4 Other Functions & Graphs

2.4.1 Exponential & Logarithmic Functions

2.4.2 Solving Equations

2.4.3 Modelling with Functions

2.5 Reciprocal & Rational Functions

2.5.1 Reciprocal & Rational Functions

2.6 Transformations of Graphs

2.6.1 Translations of Graphs

2.6.2 Reflections of Graphs

2.6.3 Stretches Graphs

2.6.4 Composite Transformations of Graphs

2.7 Polynomial Functions

2.7.1 Factor & Remainder Theorem

2.7.2 Polynomial Division

2.7.3 Polynomial Functions

2.7.4 Roots of Polynomials

2.8 Inequalities

2.8.1 Solving Inequalities Graphically

2.8.2 Polynomial Inequalities

2.9 Further Functions & Graphs