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AHL 2.15—Solutions of inequalities - IB Questionbank | RevisionDojo

Practice AHL 2.15—Solutions of inequalities with authentic IB Mathematics Analysis and Approaches (AA) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like functions and equations, calculus, complex numbers, sequences and series, and probability and statistics. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

HLPaper 1

Solve the inequality:

$\sqrt{3 x+4} \leq 4-x, x \geq-\frac{4}{3}$ .

[6]

Question 2

HLPaper 1

Consider the function $f(x)=\frac{3 x+1}{x-2}$ , where $x \in \mathbb{R}, x \neq 2$ .

Find the intercepts of the graph of $y=f(x)$ .

[2]

Solve the inequality $\frac{3 x+1}{x-2} \geq 3$ .

[5]

Question 3

HLPaper 1

Consider the function $f(x)=\frac{x+2}{x-3}$ , where $x \in \mathbb{R}, x \neq 3$ .

Write down the equations of the vertical and horizontal asymptotes of the graph of $y=f(x) .[2]$

[2]

Sketch the graph of $y=f(x)$ for $-5 \leq x \leq 8$ , clearly indicating the asymptotes and intercepts with the axes.

[3]

Solve the inequality $\frac{x+2}{x-3}<1$ .

[3]

Question 4

HLPaper 1

Consider the function $f(x)=\ln (2 x-1)-\sin (x)$ , where $x>\frac{1}{2}$ .

Sketch the graph of $y=f(x)$ for $0.5<x \leq 4$ , showing the intercepts and the behavior at $x=\frac{1}{2}$ .

[3]

Solve the inequality $\ln (2 x-1)<\sin (x)$ . [

[4]

Question 5

HLPaper 2

Consider the functions $f(x)=\frac{\sin (x)}{x-\pi}$ and $g(x)=\frac{\cos (x)}{x+\pi}$ , where $x \in \mathbb{R}, x \neq \pi, x \neq-\pi$ .

Sketch the graphs of $y=f(x)$ and $y=g(x)$ for $-2 \pi \leq x \leq 2 \pi$ , showing asymptotes and intercepts.

[4]

Solve the inequality $f(x)>g(x)$ for $-2 \pi<x<2 \pi$ .

[8]

Question 6

HLPaper 1

Let $f(x)=\frac{x^{2}+px+q}{x-3},\, x\ne3.$

Express $f(x)=x+p+3+\dfrac{q+3p+9}{x-3}$ and state the slant asymptote.

[3]

Prove $f$ is neither even nor odd for any $p,q$ .

[4]

Parameters for exactly one $x$ -intercept; give its coordinate.

[7]

For $q+3p+9\ne0$ and $\Delta>0$ , solve $f(x)\ge x+p+3$ .

[6]

Question 7

HLPaper 2

Consider $f(x)=\frac{x^3+4x^2+x+\lambda}{x^2 - x - 6},\qquad x\in\mathbb{R}\setminus\{-2,3\},\ \lambda\in\mathbb{R}.$

Show the vertical asymptotes are independent of $\lambda$ and find them.

[2]

Write $f(x)=q(x)+\dfrac{r(x)}{x^2-x-6}$ with $q$ linear and $\deg r\le1$ .

[4]

Deduce the oblique asymptote.

[2]

Find the intersection point(s) of the graph with its oblique asymptote.

[4]

Solve $f(x)\ge x+5$ for $\lambda=-30$ .

[6]

Question 8

HLPaper 1

Let $f(x)=\frac{x^{2}+px+q}{x+3},\,x\ne-3.$

Divide to obtain $f(x)=Q(x)+\dfrac{R(x)}{x+3}$ and the slant asymptote.

[3]

Show $f$ is neither even nor odd for any $p,q$ .

[4]

Find all $(p,q)$ for exactly one $x$ -intercept; give it.

[7]

For $q-3p+9\ne0$ and $\Delta>0$ , solve $f(x)\ge x+p-3$ .

[6]

Question 9

HLPaper 2

Define $f(x)=\frac{x^3+2x^2+6x+\lambda}{x^2-2x-3},\qquad x\neq-1,3.$

Find the vertical asymptotes and justify independence from $\lambda$ .

[2]

Express $f(x)=q(x)+\dfrac{r(x)}{x^2-2x-3}$ with $q$ linear.

[4]

State the slant asymptote.

[2]

Find the intersection with $y=x+4$ .

[4]

Solve $f(x)\ge x+4$ for $\lambda=-12$ .

[6]

Question 10

HLPaper 2

Given $f(x)=\frac{ax+2}{-x+a},\quad a\in\mathbb R,\ x\ne a.$

Find $f^{-1}(x)$ and its domain.

[3]

Determine when $f$ is self-inverse.

[2]

For $a=0$ , find fixed points and solve $f(x)\ge x$ .

[4]

For $a=1$ , solve $f(x)=\sqrt{x+5}$ numerically to 3 s.f., stating the number of solutions.

[3]

AHL 2.15—Solutions of inequalities Questionbank