C4.1 Relate equation solutions to inequality solutions

Ontario Grade 12 Mathematics

Polynomial and Rational Functions

How equality points help identify where an expression is positive or negative

An equation asks where two expressions are equal. An inequality asks where one expression is greater than, less than, or possibly equal to another. These questions are connected: the equality solutions often mark the points where the inequality can change from true to false. In this lesson, you will use those points and test values to describe all solutions of a related inequality.

What you will learn

Explain how solutions of an equation can help solve a related inequality.
Use equation solutions to divide the number line into intervals.
Determine which intervals satisfy a strict or inclusive inequality.

1. Prerequisite bridge: equations and inequalities

A solution is a value that makes a statement true. For example, the equation

x-4=0

has the solution

x=4

. The inequality

x-4>0

is true for every value greater than

4

.

The symbols

>

and

<

mean greater than and less than. The symbols

\geq

and

\leq

include equality. An equation may have a few solutions, while an inequality can have a whole interval of solutions.

When an inequality involves a function or expression, compare it with zero. For example,

f(x)>0

asks where the function's values are positive. The related equation

f(x)=0

identifies inputs where the function's value is zero.

f(x)=0

An equation solution makes the two sides equal.
An inequality solution makes the comparison true.
For $f(x)>0$ , first consider where $f(x)=0$ .

2. Why equation solutions matter

Suppose an expression is continuous, meaning its graph has no breaks. Its graph can move from above the horizontal axis to below it only by meeting the axis. The points where the graph meets the axis solve

f(x)=0

. Those points can therefore mark boundaries between intervals where

f(x)

is positive and intervals where it is negative.

The equation solutions do not, by themselves, tell you which intervals satisfy the inequality. You must also determine the sign of the expression on each interval. A test value from an interval is a simple way to do that when the expression is defined there.

For a strict inequality such as

f(x)>0

, points where

f(x)=0

are not included. For an inclusive inequality such as

f(x)\geq 0

, those zero points are included. The same distinction applies to

<

and

\leq

.

f(x)>0

Equation solutions can divide the number line into intervals.
Check the sign of the expression in each interval.
Include zero points only when the inequality allows equality.

3. Representing the connection

Consider the expression

(x-1)(x-3)

. Its related equation has solutions

x=1

and

x=3

. These values divide the number line into three intervals: values below

1

, values between

1

and

3

, and values above

3

.

A sign table records the sign of each factor in those intervals. When both factors have the same sign, their product is positive. When their signs differ, their product is negative. The table shows how the equation solutions connect to the inequality solutions.

This approach is useful when a graph is not supplied. A graph can also show the same information: where it lies above the horizontal axis,

f(x)>0

; where it lies below,

f(x)<0

; and where it meets the axis,

f(x)=0

.

(x-1)(x-3)=0

The equation's roots mark interval boundaries.
A sign table or graph identifies where the expression is positive or negative.
An inequality answer is usually a set of intervals, sometimes including boundary points.

4. A reliable method

To solve an inequality such as

f(x)>0

, first solve the related equation

f(x)=0

. Place its real solutions in order on a number line. They divide the number line into intervals.

Choose one test value from each interval and substitute it into the expression. Record whether the result is positive or negative. Select the intervals whose signs match the inequality. Finally, decide whether to include the equation solutions: strict inequalities exclude them, while inclusive inequalities include them when they are allowed in the expression.

Always check that the expression is defined at any proposed endpoint. The method relies on knowing where the expression is zero and checking its sign between those points; it does not mean every equation solution automatically satisfies the inequality.

f(x)≥ 0

Solve the related equation first.
Test one value in each interval.
Apply the strict or inclusive endpoint rule.

Signs of the factors and product

Interval	Sign of $x-1$	Sign of $x-3$	Sign of product
$x<1$	Negative	Negative	Positive
$1<x<3$	Positive	Negative	Negative
$x>3$	Positive	Positive	Positive

Worked example

Use roots to solve a quadratic inequality

Solve

(x-1)(x-3)\geq 0

and explain how the related equation helps.

Find the equality points
Start with the related equation. A product is zero when at least one factor is zero, so these values are the boundaries for checking the inequality.
$(x-1)(x-3)=0 \Rightarrow x=1, x=3$
Make intervals
The boundary values divide the number line into three intervals. Choose one convenient test value from each interval: $0$ , $2$ , and $4$ .
$(-\infty,1), (1,3), (3,\infty)$
Check the signs
At $x=0$ , both factors are negative, so their product is positive. At $x=2$ , one factor is positive and the other is negative, so the product is negative. At $x=4$ , both factors are positive, so the product is positive.
$(+)(+\text{ product}), (-)(+\text{ product}), (+)(+\text{ product})$
Choose the valid values
The inequality asks for a product greater than or equal to zero. Keep the two positive intervals and include $1$ and $3$ , where the product equals zero. x≤ 1 or x≥ 3

Answer: The solution is

x\leq 1

or

x\geq 3

.

Check: At

x=0

and

x=4

, the product is positive. At each endpoint,

x=1

or

x=3

, it is zero, which is allowed by

\geq

.

Common mistakes and how to avoid them

Reporting only the solutions of the related equation as the inequality answer.

Correction: The equation solutions mark boundaries. Test intervals to see where the inequality is true.

Including a boundary point for a strict inequality such as

f(x)>0

.

Correction: At a boundary point where

f(x)=0

, a strict inequality is false. Exclude that point.

Forgetting to include boundary points for an inclusive inequality.

Correction: For

f(x)\geq 0

or

f(x)\leq 0

, include a boundary point where

f(x)=0

, provided the expression is defined there.

Lesson summary

Solve the related equation to find zero points.
Use those points to divide the number line into intervals.
Check the sign in each interval with a test value or a graph.
Choose intervals that satisfy the inequality and include endpoints only when equality is allowed.

Check your understanding

Question 1

The related equation for

(x+2)(x-4)>0

has solutions

x=-2

and

x=4

. Which set solves the inequality?

$x<-2$ or $x>4$
$-2<x<4$
$x\leq -2$ or $x\geq 4$
All real values

Show answer and explanation

x<-2

or

x>4

Outside the roots, the two factors have the same sign, so their product is positive. The inequality is strict, so the zero points are excluded.

Question 2

If a graph of

f

meets the horizontal axis at

x=5

, what does that tell you about the equation and an inequality?

$x=5$ solves $f(x)=0$ , but whether it solves $f(x)\geq 0$ depends on the inequality and whether the function is defined there.
$x=5$ must solve every inequality involving $f$ .
The graph must be above the axis on both sides of $5$ .
$x=5$ cannot be part of an inequality solution.

Show answer and explanation

x=5

solves

f(x)=0

, but whether it solves

f(x)\geq 0

depends on the inequality and whether the function is defined there.

An axis intersection means

f(5)=0

. That point is included in an inequality with equality allowed, but not in a strict positive or negative inequality.

Key terms

Solution: A value that makes an equation or inequality true.
Root or zero: An input value that makes a function equal to zero.
Interval: A continuous range of numbers between boundary values.
Strict inequality: An inequality using $<$ or $>$ , which does not include equality.
Inclusive inequality: An inequality using $\leq$ or $\geq$ , which allows equality.

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C4.1 · Relate equation solutions to inequality solutions