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

Consider the functions $f\left(x\right)=3x^2+x-2$ and $g\left(x\right)=-2x^2+3x+5$.

Sketch the graph of the function $f\left(x\right)$ on the axes provided, labelling the vertex as well as the $x$- and $y$-intercepts.

Solve the inequality $f\left(x\right)<g\left(x\right)$.

Solve the inequality $5x^2-8x-48\ge 2x^2+4x-12.$

Consider the inequality $\frac{x^2-3x-10}{x-1}<0$.

Explain why you need to consider the cases $x<1$, $x=1$ and $x>1$ separately when rearranging the inequality to find a solution.

Solve the inequality.

The functions $f$ and $g$ are defined such that $f\left(x\right)=\frac{x+4}{2x-1}$ and $g\left(x\right)=2x-4$.

Given that $f$ has the largest possible domain, state the domain and range of $f$.

Solve the inequality $f\left(x\right)\le g\left(x\right)$.

Consider the function $f\left(x\right)=-2\sin x$ in the interval $-2\pi \le x\le 2\pi$.

Sketch a graph of the function over the given interval on the axes provided, labelling all $x$-intercepts as well as local minima and maxima.

Solve the inequality $f\left(x\right)>1$.

Consider the functions $f\left(x\right)=x^2-9+\frac{4}{x}$ and $g\left(x\right)=-x+5$.

Sketch the graphs of $f\left(x\right)$ and $g\left(x\right)$, clearly labelling any points of intersection or asymptotes.

Hence or otherwise, solve the inequality $f\left(x\right)\ge g\left(x\right)$.

Consider two functions, $f\left(x\right)=\ln \left(x+3\right)+4$ and $g\left(x\right)={\mathrm{e}}^{x-3}$.

Sketch both functions on the axes below, clearly labelling the asymptotes and points of intersection.

Hence or otherwise, solve the inequality $f\left(x\right)\ge g\left(x\right)$.

Consider the polynomial $q\left(x\right)=x^3-8x^2+19x-12.$ 

Given that $(x-4)$ is a factor of $q\left(x\right)$, determine the $x$-intercepts of $q\left(x\right).$

Hence or otherwise, solve the inequality $x^3+19x\le 8x^2+12$.

Consider the two functions $f\left(x\right)=2\sin 2x$ and $g\left(x\right)=\cos x$, both having the domain $0\le x\le 2\pi$.

Solve the inequality $f\left(x\right)\ge g\left(x\right)$.

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

$\left|\sin 2x\right|\ge 1-\frac{x}{360}$

for $0\le x\le 360$, where $x$ is measured in degrees.

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

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 $0<x<\frac{1}{b}$.

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.

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) $2\ln \left(x-4\right)\le \ln \left(x-4\right)$

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

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 part (b) for 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 $x^3-10x^2+32x-32\le {\left(x-4\right)}^2$.

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 $f\left(x\right)=x^2-6ax+b+10$ 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 $b$.

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

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

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

Consider the graphs with equations

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

Explain why the two graphs do not intersect.

Consider the graphs with equations

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

(i) Find the coordinates of any points of intersection 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 $f\left(x\right)>g\left(x\right)>h\left(x\right)$.

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

Find the values of $a$ and $b$.

Determine the set of values of $x$ 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

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

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

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

The region $R$ is defined 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.

Find the exact values for $x$ such that

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