THE TEST POINT METHOD FOR SENTENCES LIKE ‘$f(x) \gt 0$’

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 are a few KEY IDEAS before we get started:

RE-ARRANGING A SENTENCE TO COMPARE TO ZERO
Every equation/inequality can be written with zero on the right-hand side.
If there are any nonzero terms on the right initially, just use the Addition Property of Equality (which also works for inequalities) to subtract them from both sides.
Thus, we can always get an equivalent sentence in one of these forms: $$ f(x) = 0 \qquad\qquad f(x) > 0 \qquad\qquad f(x) \ge 0 \qquad\qquad f(x) < 0 \qquad\qquad f(x) \le 0 $$

For example, subtracting $\,3\,$ from both sides of ‘$\,x^2 \ge 3\,$’ gives the equivalent inequality ‘$\,x^2 - 3\ge 0\,$’.

ONLY TWO PLACES WHERE A FUNCTION CAN CHANGE ITS SIGN: ZEROES and BREAKS
There are only two types of places where the graph of a function $\,f\,$ can go from above to below the $x$-axis (or vice versa):
where $\,f(x) = 0$
(at a zero of the function)
SIGN CAN CHANGE
AT A ZERO
where there's a break
in the graph of $\,f$
SIGN CAN CHANGE
AT A BREAK
In other words, there are only two types of places where a function can change its sign (from positive to negative, or from negative to positive): at a zero, or at a break.

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

NO SIGN CHANGE AT THIS ZERO;
NO SIGN CHANGE AT THIS BREAK

The Test Point Method for sentences like ‘$f(x) \ge 0$’:
complete solution with full discussion

In this first example, ‘$\,x^2 \ge 3\,$’ is solved using the ‘compare to zero’ (or ‘one-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.

YOU WRITE THIS DOWNCOMMENTS
$x^2 \ge 3$original sentence
$x^2 - 3\ge 0$ rewrite the inequality with zero on the right-hand side;
since the inequality symbol is ‘$\,\ge\,$’,
we need to determine where the graph of $\,x^2 - 3\,$
lies on ($\,=\,$) or above ($\,\gt\,$) the $x$-axis
$f(x) := x^2 - 3$ if desired, name the function on the left-hand side $\,f\,$,
so it can be easily referred to in later steps;
recall that ‘$:=$’ means ‘equals, by definition’
$x^2 - 3 = 0$
$x^2 = 3$
$x = \pm\sqrt 3$

(no breaks in graph of $\,f\,$)
identify the candidates for sign changes for $\,f\,$:
  • where $\,f(x)\,$ is zero
  • breaks in the graph of $f$
  SIGN OF $\,f(x)\,$
  • mark the candidates from the previous step on a number line which is labeled ‘SIGN OF $\,f(x)\,$’
  • mark zeroes with the tick mark ‘$\,z\,$ ’
  • test each subinterval to see if $\,f(x)\,$ is $\,+\,$ or $\,-\,$;
    since $\,\sqrt 3\approx 1.7\,$, test points $\,-2\,$ and $\,2\,$ are easy
  • If the graph of $\,f\,$ is easy to obtain, then draw it in;
    in this case, you don't even need the test points!
    Graph above the $x$-axis? Then $\,f(x)\,$ is positive.
    Graph below the $x$-axis? Then $\,f(x)\,$ is negative.
    In this test point sketch, both the test points and graph are shown.
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.
For this problem, since the inequality symbol is ‘$\,\ge\,$’, we need to read off all values corresponding to plus signs and zeroes.

A second example: a function with both a break and a zero

Here is a second example, where there is both a break in the graph and a zero.
Only the in-a-nutshell version of the solution is given.

YOU WRITE THIS DOWNCOMMENTS
$\displaystyle \frac 1x \lt 2$original sentence
$\displaystyle \frac 1x - 2 \lt 0$ rewrite the inequality with zero on the right-hand side;
since the inequality symbol is ‘$\,\lt\ $’,
we need to determine where the graph of $\displaystyle\,\frac 1x - 2\,$
lies below the $x$-axis
$\displaystyle f(x) := \frac 1x - 2$ if desired, name the function on the left-hand side $\,f\,$,
so it 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 $\,x\ne 0\,$:

$\displaystyle \frac 1x - 2 = 0$

$\displaystyle \frac 1x = 2$

$x = \frac 12$   (take reciprocals of both sides: if two numbers are equal, so are their reciprocals)
identify the candidates for sign changes for $\,f\,$:
  • where $\,f(x)\,$ is zero
  • breaks in the graph of $f$
  SIGN OF $\,f(x)\,$
  • mark the candidates from the previous step on a number line which is labeled ‘SIGN OF $\,f(x)\,$’
  • mark zeroes with the tick mark ‘$\,z\,$ ’
  • mark breaks with the tick mark ‘$\,B\,$ ’
  • test each subinterval to see if $\,f(x)\,$ is $\,+\,$ or $\,-\,$
  • If the graph of $\,f\,$ is easy to obtain, then draw it in;
    in this case, you don't even need the test points!
    Graph above the $x$-axis? Then $\,f(x)\,$ is positive.
    Graph below the $x$-axis? Then $\,f(x)\,$ is negative.
    In this test point sketch, both the test points and graph are shown.
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.
For this problem, since the inequality symbol is ‘$\,\lt\,$’, we need to read off all values corresponding to minus signs.

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 for Sentences like ‘$\,f(x) \gt g(x)\,$’
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, dashed green)
  • graph of $\,y = 3\,$ (the right side of the inequality, dashed purple)
  • graph of $\,y = x^2 - 3\,$ (the function used for the ‘compare with zero’ test point method; solid red)
In this example, you are finding the values of $\,x\,$ where the green graph lies on or above the purple graph.
Equivalently, you are finding where the red graph lies on or above the $x$-axis.
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.)