THE TEST POINT METHOD FOR SENTENCES LIKE ‘$f(x) \gt g(x)$’

In Precalculus, it's essential that you can easily and efficiently solve sentences like   ‘[beautiful math coming... please be patient]$\,\frac{3x}{2}-1 \ge \frac 15 - 7x\,$’   and   ‘[beautiful math coming... please be patient]$\,x^2 \ge 3\,$’ .

The first sentence is an example of a linear inequality in one variable; a prior web exercise covers this type of sentence.
For linear inequalities, the variable appears in the simplest possible way—all you have are numbers, times $\,x\,$ to the first power
(i.e., terms of the form $\,kx\,$, where $\,k\,$ is a real number).

The second sentence is an example of a nonlinear inequality in one variable.
In nonlinear sentences, the variable appears in a more complicated way—perhaps you have an $\,x^2\,$ (or higher power), or $\,|x|\,$, or $\,\sin x\,$.
For solving nonlinear inequalities, more advanced tools are needed, which are discussed in detail in this web exercise and the next.

There are two basic methods for solving nonlinear inequalities in one variable.
These two methods were introduced in the prior web exercise, Solving Nonlinear Inequalities in One Variable (Introduction).
Both are called ‘test point methods’, because they involve identifying important intervals, and then ‘testing’ a number from each of these intervals.
The two methods are:

There is a KEY IDEA before we get started:

For the ‘two-function’ test point method, it is necessary to understand the interaction between two graphs.
We need to know where one of the graphs lies above (or below) the other graph.
As illustrated below, there are only two types of places where a graph can change from being (say)
above another graph to below it—at an intersection point, or at a break in one of the graphs:

ONLY TWO PLACES WHERE THE ABOVE/BELOW RELATIONSHIP BETWEEN TWO GRAPHS CAN CHANGE
There are only two types of places where the graph of one function, $\,f\,$, can go from above to below the graph of another function, $\,g\,$ (or vice versa):
at an intersection point
(where $\,f(x) = g(x)\,$)
red is above purple
to left of INT

red is below purple
to right of INT
where there's a break
in the graph of $\,f\,$
or the graph of $\,g\,$
red is above purple
to the left of the break

red is below purple
to the right of the break

It's important to note that there doesn't have to be an above/below change at an intersection point,
and there doesn't have to be an above/below change at a break.
These are just the candidates for places where an above/below change can occur.
That is, the following implications are true:

NO ABOVE/BELOW CHANGE AT THIS INTERSECTION POINT;
NO ABOVE/BELOW CHANGE AT THIS BREAK

The Test Point Method for sentences like ‘$f(x) \ge g(x)$’:
complete solution with full discussion

In this first example, ‘$\,x^2 \ge 3\,$’ is solved using the ‘truth’ (or ‘two-function’) method.
A full discussion accompanies this first solution; after, an in-a-nutshell version of the solution is given.

The in-a-nutshell version of the solution

Here is the ‘in-a-nutshell’ version of the previous solution.
This is probably the minimum amount of work that a teacher would want you to show.
In the exercises in this section, you will be expected to show all of these steps.
The comments are for your information, and do not need to be included in your solutions.

$x^2 \ge 3$original sentence
$f(x) := x^2$,   $g(x) := 3$ if desired, define functions $\,f\,$ (the left-hand side)
and $\,g\,$ (the right-hand side), so they can be easily referred to in later steps
$x^2 = 3$
$x = \pm\sqrt 3$

(no breaks in either graph)
find where $\,f(x)\,$ and $\,g(x)\,$ are equal (intersection points);
find any breaks in the graphs of $\,f\,$ and $\,g\,$
  TRUTH OF ‘$\,x^2 \ge 3\,$’
  • mark intersection points and breaks on a number line, which is labeled ‘TRUTH OF ‘$\,x^2 \ge 3\,$’
  • mark intersection points with the tick mark $\,i\,$
  • using easy test points from each subinterval, check if ‘$\,x^2 \ge 3\,$’ is TRUE or FALSE, and mark accordingly
solution set: $(-\infty,-\sqrt 3] \cup [\sqrt 3,\infty)$

sentence form of solution:   $x\le -\sqrt 3\ \ \text{ or }\ \ x \ge\sqrt 3$
read off the solution set, using correct interval notation;
or, give the sentence form of the solution

A second example, involving both a break and an intersection point

Here is a second example, where there is both a break in one of the graphs, and an intersection point.
Only the in-a-nutshell version of the solution is given.

YOU WRITE THIS DOWNCOMMENTS
$\displaystyle \frac 1x \lt 2$original sentence
$\displaystyle f(x) := \frac 1x\,$,     $g(x) := 2$ if desired, name the function on the left-hand side $\,f\,$
and the function on the right-hand side $\,g\,$,
so they can be easily referred to in later steps;
recall that ‘$:=$’ means ‘equals, by definition’
When $\,x = 0\,$ there is a break, since division by zero is not allowed.

For $\,x\ne 0\,$:

$\displaystyle \frac 1x = 2$

$x = \frac 12$   (take reciprocals of both sides: if two numbers are equal, so are their reciprocals)
identify the candidates for above/below relationship changes:
  • where $\,f(x) = g(x)\,$
  • breaks in the graph of $\,f\,$ or $\,g$
  TRUTH OF ‘$\displaystyle\, \frac 1x \lt 2\,$’
  • mark the candidates from the previous step on a number line which is labeled ‘TRUTH OF $\,\frac 1x \lt 2\,$’
  • mark intersection points with the tick mark ‘$\,i\,$ ’
  • mark breaks with the tick mark ‘$\,B\,$ ’
  • test each subinterval to see if the sentence is true or false
solution set: $(-\infty,0) \cup (\frac 12,\infty)$

sentence form of solution:   $x < 0\ \ \text{ or }\ \ x > \frac 12$
Read off the solution set, using correct interval notation;
or, give the sentence form of the solution.

Using WolframAlpha to solve nonlinear inequalities

Just for fun, jump up to wolframalpha.com and key in the two examples explored in this web exercise:

Voila!

Master the ideas from this section
by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:
The Test Point Method ‘in a Nutshell’ and Additional Practice
On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.

For graphical insight into the solution set, a graph is sometimes displayed.
For example, the inequality ‘$\,x^2 \ge 3\,$’ is optionally accompanied by:

  • graph of $\,y = x^2\,$ (the left side of the inequality, green)
  • graph of $\,y = 3\,$ (the right side of the inequality, purple)
In this example, you are finding the values of $\,x\,$ where the green graph lies on or above the purple graph.
Click the “show/hide graph” button if you prefer not to see the graph.

PROBLEM TYPES:
1 2 3 4 5 6 7
AVAILABLE MASTERED IN PROGRESS

(MAX is 7; there are 7 different problem types.)