Inequalities (DP IB Analysis & Approaches (AA): HL): Exam Questions

Sketch the graph of the function $f\left(x\right)=x\left(x-2\right){\left(x-4\right)}^2$.
Mark your sketch clearly with the $x$-coordinates of the $x$-axis intercepts.

Write down the solution to the inequality $f\left(x\right)\le 0$.

Briefly explain how the graph shows that there are no real solutions to the inequality $f\left(x\right)+10\le 0$.

Consider the function defined by $g\left(x\right)={\mathrm{e}}^{-\left|x\right|}$.

(i) Sketch the graph of $y=g\left(x\right)$.

(ii) Solve the inequality $g\left(x\right)\ge 0.5$ using exact values.

Consider the function $f\left(x\right)=6x^3-19x^2+16x-4.$

Fully factorise $f\left(x\right)$.

Solve

(i) $f\left(x\right)<0$

(ii) $f\left(2x\right)<0$

(iii) $f\left(x-3\right)<0$

(iv) $\left|f\left(x-3\right)\right|\le 0$

Find the values of $k$ such that the equation $kx=k{x}^2+k-2$ has real solutions and the equation $\left(4k-3\right)x^2+2kx+1=0$ has no real solutions.

Giving answers to three significant figures, find the set of values of $x$ that satisfy

$\left|\sin \left(2x^{°}\right)\right|\ge 1-\frac{x}{360}$ 

for $0\le x\le 360.$

Explain how your answer to part (a) would differ if the domain of $x$ was changed from $0\le x\le 360$ to $x\in \mathrm{ℝ}$.

Write down the set of values of $x$ for which $e^{2x}\le e^x$.

Find the set of values of $x$ for which 

          (i)    $e^{2\left(x-4\right)}\le e^{x-4}$ 

          (ii)    $e^{5\left(x-4\right)}\ge e^{-\left(x-4\right)}$

Find the set of values of $x$ for which

(i) $2\ln (x-4)\le \ln (x-4)$

(ii) $-\ln (x-4)\ge 5\ln (x-4).$

Consider the functions

$f\left(x\right)=\frac{a}{a-x}$  and  $g\left(x\right)=\frac{2a}{x-2a}$ 

where $a$ is a real constant such that $a\ne 0.$ 

In terms of the constant $a$, find 

          (i)     the values of $x$ for which $f\left(x\right)$ and $g\left(x\right)$ are undefined, 

          (ii)    the $x$-coordinate of any intersections between the graphs of $y=f\left(x\right)$ and $y=g\left(x\right)$.

In the case $a>0$, find the set of values of $x$ in terms of $a$ for which 

          (i)     $f\left(x\right)>g\left(x\right)$, 

          (ii)    $f\left(x\right)<g\left(x\right)$.

Repeat questions (b) (i) and (ii) in the case $a<0$.

$f\left(x\right)=x^3-11x^2+40x-48$

(i) Show that $x=3$ is a root of the function .

(ii) Hence fully factorise $f\left(x\right)$.

Solve the inequality $x^3-10x^2+32x-32\le (x-4{)}^2.$

Solve the inequality $\left(x+a\right){\left(x+b\right)}^2>0$, where $a$ and $b$ are constants such that $a>b>0$.  Give your answers in terms of $a$ and $b$.

Consider the functions defined by  $f\left(x\right)=x^2-a$ and $g\left(x\right)=a-x^2$,  where $a$ is a positive constant.
Solve the inequality $f\left(x\right)<g\left(x\right)$,  giving your answer in terms of $a$.

Consider the functions defined by  $p\left(x\right)=x^3-b$ and  $q\left(x\right)=b-x^3$,  where $b$ is a real constant.
Solve the inequality $p\left(x\right)<q\left(x\right)$,  giving your answer in terms of $b$.

Consider the functions defined by $h\left(x\right)=x^n-m$ and $j\left(x\right)=m-x^n$,  where $m>0$ and $n\in {\mathrm{\mathbb{Z}}}^{+}$.
Write down, in terms of $m$ and $n$, the solution to the inequality $h\left(x\right)<j\left(x\right)$when

(i) $n$ is even,

(ii) $n$ is odd.

Consider the functions defined by $f\left(x\right)=a+\ln bx$ and $g\left(x\right)=a-\ln bx$ where $a$ and $b$ are positive constants.
Show that

$f\left(x\right)<g\left(x\right)$ for  $0<x<\frac{1}{b}$.   .

Consider the functions defined by $f\left(x\right)=x^2-6ax+b+10$ and $g\left(x\right)=ax+2b+3,$where $a,b\in \mathrm{\mathbb{Z}}$.

Given that $f\left(x\right)\le g\left(x\right)$ only for $2\le x\le 5,$ find the values of a and b.

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

Find the values of a and b.

Determine the set of values of that satisfy

(i) $f\left(x\right)\ge 0$,

(ii) $f\left(-x\right)\ge 0,$

(iii) $-f\left(x\right)<0.$

Determine the smallest positive value $k$ such that the solution to the inequality $f\left(x\right)\le k$  is a single interval.

The function $f$ is such that 

 $f\left(x\right)\ge 0\mathrm{for}x\le 3\mathrm{and}\mathrm{for}4\le x\le 5,$

$f\left(x\right)\le 0\mathrm{for}3\le x\le 4\mathrm{and}\mathrm{for}x\ge 5.$

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

Sketch the graph of $y=f\left(x\right)$ where

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

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

Find the values of  $x$ that satisfy

(i) $f\left(x\right)\ge 0.$

(ii) $f\left(\right|x\left|\right)\ge 0.$

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

$y\ge 1,$

$y\le 2x+8,$

$x+y\le 10.$

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

Consider the graphs with equations

$y=\frac{\left(x+4\right)\left(x-1\right)}{x-1}\mathrm{and}y=6-x$. 

Explain why the two graphs do not intersect.

Consider the graphs with equations

$y=\frac{\left(x-6\right){\left(x-1\right)}^2}{x-1}\mathrm{and}y=\left(8-x\right)\left(x-1\right)$.

(i) Find the coordinates of any points of intersections between the two graphs.

(ii) Hence, or otherwise, solve the inequality $\frac{\left(x-6\right){\left(x-1\right)}^2}{x-1}\le \left(8-x\right)\left(x-1\right).$

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

On the same diagram, sketch the graphs of $f,g$and $h$.

Find the set of values of $x$ which satisfy the inequality $f\left(x\right)>g\left(x\right)>h\left(x\right)$.

Find the exact values for $x$ such that

$\frac{x}{\left(x+2\right)\left(x-3\right)}\ge x$