C4.2 Solve polynomial and rational inequalities graphically

Ontario Grade 12 Mathematics

Polynomial and Rational Functions

Use a function’s position relative to the horizontal axis to find the solution

An equation asks when two expressions have the same value. An inequality asks when one expression is greater or less than another. A graph makes that comparison visible. If a function’s graph is above the horizontal axis, its output is positive. If it is below the axis, its output is negative. This lesson uses those observations to solve polynomial and rational inequalities. You will identify important inputs, read the graph on each interval, and decide which endpoints belong in the solution.

What you will learn

Interpret polynomial and rational inequalities by graphing a function.
Use zeros and undefined inputs to identify intervals to inspect.
Choose solution endpoints correctly for strict and inclusive inequalities.
Check graph-based conclusions with plotted values when useful.

1. Prerequisite bridge: expressions and zeros

A polynomial is an expression made from terms with non-negative whole-number powers of a variable. For example,

x^2-4

is a polynomial. A rational expression is a fraction whose numerator and denominator are polynomials, such as

\frac{x+1}{x-2}

.

A function gives an output for an input. A zero is an input that makes the output equal to zero. On a graph, a zero appears as an

x

-intercept: a point where the graph meets the horizontal axis. A rational function is undefined when its denominator is zero. It has no graph point at that input.

These ideas connect inequalities to graphs. Solving

f(x)>0

means finding inputs where the graph of

y=f(x)

is above the horizontal axis. Solving

f(x)<0

means finding inputs where it is below. For

f(x)\geq 0

or

f(x)\leq 0

, points on the axis can also be included, but only when the function is defined there.

If an inequality compares two expressions, first put them on one side. For example,

p(x)>q(x)

can be rewritten as

p(x)-q(x)>0

. Then graph the function made by the difference. This turns the comparison into a question about whether one graph is above or below the horizontal axis.

y=f(x)

A zero is an input where the function’s output is zero.
A rational function is undefined when its denominator is zero.
A graph above the axis has positive outputs; a graph below has negative outputs.

2. Read intervals and endpoints from a graph

Begin by marking the important inputs on the horizontal axis. For a polynomial, mark its zeros. For a rational function, mark its zeros and every input where its denominator is zero. These values divide the number line into intervals. An interval is a continuous stretch of values between marked inputs.

On each interval, inspect the graph’s position. If the graph is above the axis, that interval satisfies a positive inequality. If it is below, it satisfies a negative inequality. A graph may touch the axis and turn back, so do not assume that the sign changes at every zero. Read the graph on both sides of each marked value.

The inequality sign determines whether a zero can be included. A strict inequality, using

>

or

<

, does not include a zero, because the function’s value there is exactly zero. An inclusive inequality, using

\geq

or

\leq

, can include a zero if the function is defined at that input. A value that makes a rational function’s denominator zero is always excluded.

In interval notation, parentheses exclude endpoints and square brackets include them. For example,

(a,b)

excludes both endpoints, while

[a,b]

includes both. A mixed interval such as

[a,b)

includes

a

but excludes

b

. Infinity is never an endpoint value, so intervals that extend without bound use parentheses at infinity.

A graphing tool can show the curve, but the window and scale matter. A curve may look close to the axis without meeting it. Use intercepts to locate zeros. If you need to confirm whether an interval is above or below the axis, inspect a plotted point within that interval. f(x)>0\iff y=f(x) is above the horizontal axis

Mark zeros and denominator restrictions before selecting intervals.
Use the graph’s position on each interval to decide whether the function is positive or negative.
Include defined zeros only for inclusive inequalities; exclude every undefined input.

3. Guided example: a rational inequality

Solve

\frac{x+1}{x-2}\geq 0

graphically. Consider the graph of

y=\frac{x+1}{x-2}

. We want the parts of the graph that are on or above the horizontal axis.

The numerator is zero at

x=-1

, so the graph has an intercept there. The denominator is zero at

x=2

, so the function is undefined there. These two inputs divide the number line into three intervals: values less than

-1

, values between

-1

and

2

, and values greater than

2

.

To read the graph’s sign, check one input from each interval. At

x=-2

, the function’s output is positive. At

x=0

, it is negative. At

x=3

, it is positive. The graph is therefore above the axis on the two outer intervals and below it in the middle interval. These checks support what the graph shows.

The inequality includes equality, so include the zero at

x=-1

. Do not include

x=2

, because the rational function is undefined there. Write the answer as the union of the two intervals where the graph is positive, including the defined zero at

-1

.

\frac{x+1}{x-2}\geq 0

The numerator’s zero is an intercept; the denominator’s zero is an excluded input.
Checking one input in each interval confirms which regions are above or below the axis.
Include the defined zero for this inclusive inequality, but exclude the undefined input.

4. Apply the method to polynomial inequalities

For a polynomial inequality, graph the polynomial as a function and mark its zeros. There are no denominator restrictions to consider. Select the intervals where the graph is above or below the horizontal axis, according to the inequality. Include zeros only if the inequality allows equality.

For example, consider

x^2-4<0

. The graph of

y=x^2-4

meets the horizontal axis at

x=-2

and

x=2

. Between those intercepts, the curve is below the axis. The inequality is strict, so its solution is the open interval between the intercepts, not including either one.

A polynomial graph can meet the axis and turn around instead of crossing it. That is why the graph’s actual position on each interval matters. The same interval-reading process applies to rational graphs, with the added rule that inputs where the denominator is zero are not part of the domain and cannot be solutions.

A clear written solution should show the intervals where the graph satisfies the inequality and explain endpoint choices. When possible, check the graph’s intercepts and use plotted values to confirm the sign on each interval. This keeps the answer tied to the graph rather than to a guess about the curve.

p(x)>q(x)\iff p(x)-q(x)>0

Graph the expression after moving all terms to one side, when needed.
Choose intervals based on the graph’s position relative to the horizontal axis.
Check endpoints separately: zeros may be included, but undefined inputs may not.

Reading the graph in the guided example

Input interval or value	Graph position	Included in the solution?
$(-\infty,-1)$	Above the axis	Yes
$x=-1$	On the axis	Yes
$(-1,2)$	Below the axis	No
$x=2$	Undefined	No
$(2,\infty)$	Above the axis	Yes

Worked example

Solve a rational inequality from its graph

Solve

\frac{x+1}{x-2}\geq 0

graphically.

Mark the important inputs
The graph meets the horizontal axis when the numerator is zero. It is undefined when the denominator is zero. Mark both inputs because they divide the number line into intervals to inspect.
$x=-1,\quad x=2$
Read the graph on each interval
Graph the function and check a point from each interval. The outputs at $x=-2$ , $x=0$ , and $x=3$ show that the graph is above the axis on the outer intervals and below it between the marked inputs.
$f(-2)=\frac{1}{4},\quad f(0)=-\frac{1}{2},\quad f(3)=4$
Apply the endpoint rules
The inequality includes equality, so include the zero at $x=-1$ . The function is undefined at $x=2$ , so exclude that input. The graph satisfies the inequality on the outer intervals.
$(-\infty,-1]\cup(2,\infty)$

Answer:

(-\infty,-1]\cup(2,\infty)

Check: At

x=-1

, the expression equals zero, so it is included. At

x=2

, the denominator is zero, so it is excluded. The checked values

x=-2

and

x=3

give positive outputs, while

x=0

gives a negative output.

Common mistakes and how to avoid them

Including a zero for a strict inequality such as

f(x)>0

.

Correction: A zero gives an output of exactly zero, not a positive output. Include it only for an inclusive inequality and only when the function is defined there.

Including an input that makes a rational function’s denominator zero.

Correction: The function has no value at that input. Exclude it from the solution, whether the inequality is strict or inclusive.

Assuming the graph changes sign at every zero.

Correction: A graph can touch the axis and turn back. Read its position on each interval instead of assuming the sign changes.

Choosing intervals from a graph without checking intercepts or the scale.

Correction: Mark the zeros and undefined inputs first. Use the graph and, if needed, plotted values within the intervals to confirm the sign.

Lesson summary

Rewrite a comparison as a function compared with zero when needed.
Graph the polynomial or rational function and mark its zeros.
For a rational function, also mark inputs where the denominator is zero.
Select intervals where the graph is above or below the axis as required.
Include defined zeros only for inclusive inequalities, and never include undefined inputs.

Check your understanding

Question 1

The graph of

y=g(x)

is below the horizontal axis for

-3<x<1

and meets the axis at both endpoints. What is the solution to

g(x)<0

on this part of the graph?

$[-3,1]$
$(-3,1)$
$(-\infty,-3)\cup(1,\infty)$
$[-3,1)$

Show answer and explanation

(-3,1)

The graph is below the axis between the intercepts. The inequality is strict, so neither endpoint is included.

Question 2

A rational function is undefined at

x=4

. Can

x=4

be included in the solution to an inequality involving that function?

Yes, if the inequality uses $\geq$ or $\leq$ .
Yes, if the graph is above the axis nearby.
No, because the function has no value there.
No, unless $x=4$ is also an intercept.

Show answer and explanation

No, because the function has no value there.

An input that makes the denominator zero is excluded because the rational function is undefined there.

Key terms

Polynomial: An expression made from terms with non-negative whole-number powers of a variable, including a constant term.
Rational expression: A fraction whose numerator and denominator are polynomial expressions.
Zero: An input that makes a function’s output equal to zero.
Interval: A continuous stretch of values on the number line.
Undefined input: An input at which a function has no value, such as an input that makes a rational function’s denominator zero.

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C4.2 · Solve polynomial and rational inequalities graphically