IB DP Maths: AA HL復習筆記2.8.1 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
- if we?move?g(x) to the LHS?we get
- 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)
The safest way?to rearrange is simply to add & subtract to move all the terms onto one side
- You can only multiply by a term if you are certain it is always positive (or always negative)
Worked Example
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