2.8 Inequalities

Question 1

Marks: 4

Consider the functions defined by $f (x) = x^{2} - 6 a x + b + 10$ and $g (x) = a x + 2 b + 3,$ where $a, b \in Z^{+}$ . Given that $f (x) \leq g (x)$ only for $2 \leq x \leq 5,$ find the values of a and b.

Key Concepts

Intersecting Graphs

Question 2a

Marks: 3

The function defined by $f (x) = x^{4} - 12 x^{3} + 46 x^{2} - 60 x + 25$ can be factorised into the form $f (x) = (x - a)^{2} (x - b)^{2},$ where $a$ and $b$ are positive integers such that $a < b$ .

(a)

Find the values of a and b.

Key Concepts

Polynomial Inequalities

Question 2b

Marks: 3

(b)

Determine the set of values of that satisfy

(i)

f (x) \geq 0

(ii)

f (- x) \geq 0,

(iii)

- f (x) < 0 .

Key Concepts

Polynomial Inequalities

Question 2c

Marks: 2

(c)

Determine the smallest positive value

k

such that the solution to the inequality

f (x) \leq k

is a single interval.

Key Concepts

Polynomial Inequalities

Question 3

Marks: 5

The function $f$ is such that

$f (x) \geq 0 for x \leq 3 and for 4 \leq x \leq 5,$

$f (x) \leq 0 for 3 \leq x \leq 4 and for x \geq 5 .$

Find a polynomial, of the lowest degree possible, that satisfies the condition $f (0) = 5$ .

Key Concepts

Polynomial Inequalities
Solving Inequalities Graphically

Question 4a

Marks: 3

(a)

Sketch the graph of

y = f (x)

where

f (x) = \frac{(x + 2) (x - 4) (x - 6)}{(x - 1) (x - 5)}

Label any intersections with the coordinate axes and state the equations of any vertical asymptotes.

Key Concepts

Graphing Functions

Question 4b

Marks: 5

(b)

Find the values of

x

that satisfy

(i)

f (x) \geq 0 .

(ii)

f (| x |) \geq 0 .

Key Concepts

Solving Inequalities Graphically

Question 5

Marks: 6

The region R is defined by the three straight lines given by the inequalities

$y \geq 1,$

$y \leq 2 x + 8,$

$x + y \leq 10 .$

The function $f$ is defined by $f (x) = 2 + \frac{1}{x - 1}$ . Find the largest domain of $f$ such that the graph of $f$ lies within the region R. Give answers as exact values where appropriate.

Key Concepts

Solving Inequalities Graphically
Linear Rational Functions & Graphs

Question 6a

Marks: 1

(a)

Consider the graphs with equations

y = \frac{(x + 4) (x - 1)}{x - 1} and y = 6 - x

Explain why the two graphs do not intersect.

Key Concepts

Solving Inequalities Graphically

Question 6b

Marks: 3

(b)

Consider the graphs with equations

y = \frac{(x - 6) {(x - 1)}^{2}}{x - 1} and y = (8 - x) (x - 1)

(i)

Find the coordinates of any points of intersections between the two graphs.

(ii)

Hence, or otherwise, solve the inequality

$\frac{(x - 6) {(x - 1)}^{2}}{x - 1} \leq (8 - x) (x - 1) .$

Key Concepts

Solving Inequalities Graphically

Question 7a

Marks: 3

Consider the functions defined by $f (x) = \sqrt{(9 - x^{2})}, g (x) = 3 - \sqrt{(9 - x^{2})}$ and $h (x) = \frac{x + 3}{2} .$ All three functions have the domain $- 3 \leq x \leq 3 .$

(a)

On the same diagram, sketch the graphs of

f, g

and

h

Key Concepts

Graphing Functions

Question 7b

Marks: 3

(b)

Find the set of values of

x

which satisfy the inequality

f (x) > g (x) > h (x)

Key Concepts

Solving Inequalities Graphically

Question 8

Marks: 6

Find the exact values for $x$ such that

$\frac{x}{(x + 2) (x - 3)} \geq x$

Key Concepts

Solving Inequalities Graphically

DP IB Maths: AA HL

Topic Questions

2.8 Inequalities

Question 1

Key Concepts

Question 2a

Key Concepts

Question 2b

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Question 2c

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Question 3

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Question 4a

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Question 4b

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Question 5

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Question 6a

Key Concepts

Question 6b

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Question 7a

Key Concepts

Question 7b

Key Concepts

Question 8

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