Introduction ​

Define $f$ by

$$f(x)=(x-2)(x+3)$$

For some choices of $x,$ the value of $f(x)$ is positive, for example:

$$\begin{array}{rl}f(5)=(5-2)(5+3)& >0\\ \\ f(-4)=(-4-2)(-4+3)& >0\end{array}$$

But for other choices of $x,$ $f(x)$ is negative, for example

$$f(1)=(1-2)(1+3)<0$$

What if we wanted to describe the set of all $x$ such that $f(x)>0$?

The easiest way is to sketch $y=f(x)$ and look for when $y>0$ (that is, when the parabola lies above the $x$-axis)

Now it is clear that $f(x)>0$ whenever $x<-3$ or $x>2.$

Over the page, we'll look at some more examples of solving this type of inequality.

Example ​

Solve the inequalities

$x^2+x-20<0$

$4-2x-3x^2<0$

$t^2-4t-6\ge 2t+1$

Exercise ​

Problem ​

Prove that the parabola

$$y=x^2-4x+7$$

lies entirely above the $x$-axis

Find the values of $x$ for which points on the parabola lie below the line

$$y=x+6$$

Problem ​

A rectangle has a width of $k+2$ and a height of $k-2,$ where $k$ is a real number.

The area of the rectangle is smaller than or equal to its perimeter.

Find the range of possible values of $k.$

Problem ​

A cubic $p$ is defined by

$$p(x)=x^3-6x^2+3x+10$$

Show that $p(2)=0.$

Hence, solve the inequality

$$p(x)\ge 0$$

Problem ​

Find the set of values of $x$ for which points on the line

$$y=x-8$$

lie inside the circle with equation

$$(x-5{)}^2+(y-3{)}^2=26$$

Problem ​

The quadratic equation

$$x^2+kx+5=0$$

has distinct, real roots.

Find the range of possible values of $k.$

Problem ​

The quadratic equation

$$x^2+(k+1)x+2k=0$$

has at least one real root.

Find the range of possible value of $k.$

Problem ​

The line $y=mx$ intersects with the parabola

$$y=x^2+1$$

Find the possible values of $m.$

Problem ​

The line

$$y=x+k$$

intersects with the circle

$$(x-2{)}^2+(y-1{)}^2=4$$

Find the possible values of $k.$