Solving Inequalities Graphically (DP IB Analysis & Approaches (AA)): Revision Note

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 

• Some functions reverse the inequality

  • Taking reciprocals of positive values

    • 

  • Taking logarithms when the base is 0 < a < 1

    • 

• The safest way to rearrange is simply to add & subtract to move all the terms onto one side