IB Mathematics AA HL Solving inequalities Study Notes | New Syllabus

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 of the form \( g(x) \ge f(x) \) : Analytically

Solving Inequalities of the form \( g(x) \ge f(x) \) : Analytically

To solve \( g(x) \ge f(x) \) analytically:

  1. Set up the inequality: \( g(x) – f(x) \ge 0 \)
  2. Form a single expression and simplify
  3. Solve the resulting equation \( g(x) – f(x) = 0 \) to find critical points
  4. Test intervals between critical points to determine where \( g(x) – f(x) \ge 0 \)

Example

Solve analytically: \( 2x + 1 \ge x^2 \)

▶️ Answer/Explanation

Rearrange the inequality:

\( 0 \ge x^2 – 2x – 1 \)

Or equivalently:

\( x^2 – 2x – 1 \le 0 \)

Roots of the quadratic:

\( x = \frac{2 \pm \sqrt{(-2)^2 – 4(1)(-1)}}{2} = \frac{2 \pm \sqrt{4 + 4}}{2} = \frac{2 \pm \sqrt{8}}{2} = \frac{2 \pm 2\sqrt{2}}{2} = 1 \pm \sqrt{2} \)

The critical points are \( 1 – \sqrt{2} \) and \( 1 + \sqrt{2} \).

Since the quadratic opens upwards, the solution is:

\( 1 – \sqrt{2} \le x \le 1 + \sqrt{2} \)

Solve analytically: \( x^3 – x^2 \ge 0 \)

▶️Answer/Explanation

Factor the expression:

\( x^2(x – 1) \ge 0 \)

The critical points are \( x = 0 \) (double root) and \( x = 1 \).

Test intervals:

  • \( x < 0 \): Choose \( x = -1 \Rightarrow (-1)^2(-1-1) = 1 \times (-2) = -2 < 0 \)
  • \( 0 < x < 1 \): Choose \( x = 0.5 \Rightarrow (0.5)^2(0.5-1) = 0.25 \times (-0.5) = -0.125 < 0 \)
  • \( x > 1 \): Choose \( x = 2 \Rightarrow 4 \times (2-1) = 4 \times 1 = 4 > 0 \)

Since \( x^2 \ge 0 \) always, we can also reason:

\( x^3 – x^2 \ge 0 \Rightarrow x^2(x – 1) \ge 0 \)

The solution is:

\( x \ge 1 \text{ or } x = 0 \)

Solving Inequalities of the form \( g(x) \ge f(x) \) : Graphically

Solving Inequalities of the form \( g(x) \ge f(x) \) : Graphically

To solve \( g(x) \ge f(x) \) graphically using technology:

  1. Graph both functions \( y = g(x) \) and \( y = f(x) \) on the same axes.
  2. Identify their points of intersection (using the GDC’s “intersect” or “analyze graph” feature).
  3. Determine the intervals where the graph of \( g(x) \) lies above or on the graph of \( f(x) \).

General GDC Steps: Solving \( g(x) \ge f(x) \)

  1. Open graphing mode on your calculator or software (e.g., TI-84: press Y=, Casio: select graph, GeoGebra: enter function).
  2. Enter functions:
    • Y1 = g(x) (your first function)
    • Y2 = f(x) (your second function)
  3. Set a suitable window:
    • Choose Xmin and Xmax to cover your domain of interest
    • Set Ymin and Ymax so you can see both functions clearly
  4. Graph both functions so you can see where they cross.
  5. Find points of intersection:
    • On TI: 2nd → TRACE → 5:intersect.
    • Select each curve, move cursor near the intersection point, press ENTER.
    • On Casio / GeoGebra / Desmos: Use Intersection or Trace tools.
  6. Determine the solution region:
    • Look at where \( g(x) \) lies above or on \( f(x) \) on the graph.
    • The solution to \( g(x) \ge f(x) \) is the x-values where this occurs.
  7. State solution in interval form or inequality form based on intersection points.

Example

Solve graphically using technology: \( e^x \ge \sin x \)

▶️ Answer/Explanation

Plot both graphs: \( y = e^x \) and \( y = \sin x \).

Use technology to find points of intersection (e.g., near \( x = -3.18 \)).

Since \( e^x \) increases and \( \sin x \) oscillates, the solution interval is:

\( x \ge \text{first intersection point} \)

Technology steps (e.g. Desmos / GDC):

  • Enter y = e^x and y = sin(x)
  • Use intersection tool to find where curves cross
  • Shade the region or identify x-values where \( e^x \ge \sin x \)
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