IB Mathematics AA HL Solving inequalities Study Notes
IB Mathematics AA HL Solving inequalities Study Notes
IB Mathematics AA HL Solving inequalities Study Notes Offer a clear explanation of Solving inequalities , including various formula, rules, exam style questions as example to explain the topics. Worked Out examples and common problem types provided here will be sufficient to cover for topic Solving inequalities.
Solving Inequalities
Solving inequalities involves understanding the behavior of various functions and their transformations, including modulus equations, dynamic graphing, and specialized cases. This section explores the techniques, examples, and international approaches to analyzing inequalities.
Graphing Transformations
The following transformations play a crucial role in solving inequalities:
- Absolute Value Functions:
- \( y = |f(x)| \): Reflects all parts of \( f(x) \) below the x-axis to above it.
- \( y = f(|x|) \): Mirrors the graph of \( f(x) \) about the y-axis.
- Reciprocal Functions:
- \( y = \frac{1}{f(x)} \): The graph has vertical asymptotes where \( f(x) = 0 \) and reflects behavior inversely.
- Horizontal and Vertical Stretches:
- \( y = f(ax+b) \): Applies a horizontal compression or stretch based on \( a \), combined with a horizontal shift \( b \).
- Squared Functions:
- \( y = [f(x)]^2 \): Amplifies values of \( f(x) \), with non-negative outputs, as squares of all values are non-negative.
Dynamic graphing tools can help visualize these transformations and understand their impact on inequalities.
Solving Modulus Equations and Inequalities
Solving modulus equations and inequalities requires splitting the solution into cases based on the definition of the modulus function:
- For \( |f(x)| > a \):
- Split into \( f(x) > a \) or \( f(x) < -a \).
- For \( |f(x)| < a \):
- Split into \( -a < f(x) < a \).
Example: Solve \( |3x \arccos(x)| > 1 \):
- Split into \( 3x \arccos(x) > 1 \) or \( 3x \arccos(x) < -1 \).
- Solve each inequality separately, considering the domain of \( \arccos(x) \), i.e., \( x \in [-1, 1] \).
Connections and Context
The study of inequalities bridges visual and analytical approaches:
- International-Mindedness:
- The Bourbaki Group: Promotes a rigorous analytical approach to mathematics, emphasizing algebraic solutions and proofs.
- The Mandelbrot Visual Approach: Focuses on intuitive, visual interpretations of mathematical concepts, especially through graphing.
- Applications in real-world modeling and optimization, such as engineering and economics.
Examples
Example 1: Absolute Value Function
Consider \( |x-2| > 3 \).
- Split into \( x-2 > 3 \) or \( x-2 < -3 \).
- Solutions: \( x > 5 \) or \( x < -1 \).
Example 2: Reciprocal Function
Consider \( \frac{1}{x+1} < 2 \).
- Rearrange: \( x+1 > \frac{1}{2} \) or \( x+1 < 0 \).
- Solution: \( x > -\frac{1}{2} \) or \( x < -1 \).
Example 3: Modulus and Square Function
Consider \( |x^2 – 4| \leq 3 \).
- Split into \( -3 \leq x^2 – 4 \leq 3 \).
- Rearrange: \( 1 \leq x^2 \leq 7 \).
- Solution: \( -\sqrt{7} \leq x \leq -1 \) or \( 1 \leq x \leq \sqrt{7} \).
Example 4: Dynamic Transformation
Use graphing software to visualize \( y = |f(x)| \) and \( y = f(|x|) \) for \( f(x) = x^2 – 4 \).
- \( y = |x^2 – 4| \): Reflects all negative portions of the graph above the x-axis.
- \( y = (|x|)^2 – 4 \): Mirrors the graph about the y-axis.
IB Mathematics AA SL Solving inequalities Exam Style Worked Out Questions
Question
Consider f (x) = 4sinx + 2.5 and \(g(x) = 4sin\left ( x-\frac{3\pi }{2} \right )+ 2.5 + q,\) where x ∈ R and q > 0. The graph of g is obtained by two transformations of the graph of f .
(a) Describe these two transformations.
The y-intercept of the graph of g is at (0, r).
(b) Given that g(x) ≥ 7, find the smallest value of r.
▶️Answer/Explanation
Ans:
(a) translation (shift) by \(\frac{3\pi }{2}\) to the right OR positive horizontal direction by \(\frac{3\pi }{2}\)
translation (shift) by q upwards OR positive vertical direction by q
Note: accept translation by \(\binom{\frac{3\pi }{2}}{q}\)
Do not accept ‘move’ for translation/shift.
(b)
minimum of \(\left ( x-\frac{3\pi }{2} \right )\) is -4 (may be seen in sketch)
-4 + 2.5 + q ≥7
q≥ 8.5 (accept q = 8.5)
substituting x = 0 and their q (= 8.5) to find r
(r =) \(4sin\left ( \frac{-3\pi }{2} \right )+2.5 + 8.5\)
4 + 2.5 + 8.5
smallest value of r is 15
Question
Consider the function f(x) = \(2^{x} – \frac{1}{2^{x}},\) x ∈ R.
(a) Show that f is an odd function.
The function g is given by g(x) = \(\frac{x-1}{x^{2}-2x-3},\) where x ∈ R, x ≠ -1 , x ≠ 3.
(b) Solve the inequality f (x) ≥ g(x).
▶️Answer/Explanation
Ans:
(a) attempt to replace x with −x
f(-x) = \(2^{-x}-\frac{1}{2^{-x}}\)
EITHER
\(=\frac{1}{2^{x}}-2^{x} = -f(x)\)
OR
\(=-\left (2^{x} \frac{1}{2^{x}} \right )\left ( =-f(x) \right )\)
Note: Award M1A0 for a graphical approach including evidence that either the graph is invariant after rotation by 180°about the origin or the graph is invariant after a reflection in the
y -axis and then in the x -axis (or vice versa).
so f is an odd function
(b) attempt to find at least one intersection point
x = -1.26686….., x = 0.177935…..,x = 3.06167
x = -1.27, x = 0.178, x = 3.06
-1.27 ≤ x < – 1,
0.178 ≤ x < 3,
x ≥ 3.06