D3.2 Solve non-algebraic equations and inequalities

Ontario Grade 12 Mathematics

Characteristics of Functions

Using graphs and numerical estimates to find and interpret solutions

Some equations and inequalities cannot be solved by isolating a variable with familiar algebraic steps. A trigonometric equation such as

\cos x=x

is one example. In this lesson, you will use graphs and numerical estimates to locate solutions and decide which values satisfy an inequality. The goal is not to find an exact value in every case. It is to find a reliable approximation and communicate what it means.

What you will learn

Explain how an equation can be solved by finding where two function graphs intersect.
Use a table and repeated estimates to approximate a solution when exact algebra is not practical.
Use graph position or sign comparisons to solve an inequality.
State solutions with appropriate attention to the domain and interval.

1. Prerequisite bridge: equations as comparisons

An equation says that two expressions have the same value. For example,

2x=6

is solved by finding the input that makes both sides equal. You already know how to check a proposed solution by substituting it into both sides.

A function assigns an output to each allowed input. Its graph shows input-output pairs: the horizontal coordinate is the input, and the vertical coordinate is the output. To solve an equation that compares two functions, look for inputs where their outputs match.

A domain is the set of inputs allowed for a function or problem. Check the domain before reporting solutions. For trigonometric functions, angle units matter: in this lesson, angles are measured in radians.

f(x)=g(x)

A solution is an input that makes an equation true.
An intersection is a point shared by two graphs.
The input coordinate of an intersection solves the equation.

2. Plain language: turn an equation into an intersection

For an equation such as

\cos x=x

, graph

y=\cos x

and

y=x

. Where the curves meet, their

y

-values are equal. The corresponding

x

-coordinate is a solution.

A graph often gives a useful first estimate, but its accuracy depends on the viewing window and scale. A numerical table can help refine the estimate. Calculate each side at nearby inputs, then compare the outputs. If one side is larger at one input and smaller at a nearby input, the solution lies between those inputs when the graphs cross there.

A numerical solution is an approximate value rather than an exact expression. Give enough decimal places for the question, and remember that rounding means the reported value may not make both sides exactly equal.

y=f(x),\quad y=g(x)

Graph both sides as separate functions.
Read the input coordinate where the graphs intersect.
Use nearby table values to improve an estimate.

3. Inequalities: compare which graph is higher

An inequality asks which inputs make one side greater or less than the other. On a graph,

f(x)>g(x)

where the graph of

f

is above the graph of

g

. For

f(x)<g(x)

, it is below.

First find the intersection inputs, since the graphs can switch which one is higher there. Then check one input in each interval between intersections. Include an endpoint only when the inequality allows equality, as in

\leq

or

\geq

, and when the endpoint is in the domain.

A sign comparison is another way to organize the same check. Subtract the right side from the left. A positive result means the left side is greater; a negative result means it is smaller. For numerical work, use enough precision near an intersection so rounding does not mislead you.

f(x)>g(x)\iff f(x)-g(x)>0

Above means greater; below means less.
Intersections mark possible changes in which function is greater.
Strict inequalities do not include equality points.

4. Guided example: estimate a trigonometric solution and inequality

We will solve and interpret the comparison between cosine and the line

y=x

on the interval from

0

to

1

radians. A sketch suggests one crossing. A table narrows down where it occurs.

Define the difference as cosine minus the input. A positive difference means the cosine graph is higher; a negative difference means it is lower. Values on either side of zero help locate the intersection.

d(x)=\cos x-x

Use the specified interval and angle units.
Refine the crossing with numerical values.
Use the sign of the difference to answer the inequality.

Comparing cosine with the input

Input $x$	$\cos x$	$\cos x-x$	Comparison
$0.70$	$0.7648$	$0.0648$	Cosine is greater
$0.75$	$0.7317$	$-0.0183$	Cosine is smaller
$0.739$	$0.7392$	$0.0002$	Very close to equal

Worked example

Cosine compared with the input

On

0\leq x\leq 1

, estimate the solution of

\cos x=x

, then solve

\cos x>x

on that interval. Use radians.

Identify the graphs
Graph $y=\cos x$ and $y=x$ on the given interval. Their crossing gives the equation solution. To study the inequality, compare the two outputs at inputs on either side of the crossing.
$y=\cos x,\quad y=x$
Bracket the crossing
At $x=0.7$ , cosine is greater than the input. At $x=0.75$ , cosine is less than the input. The crossing is therefore between these values.
$\cos(0.7)-0.7\approx 0.0648,\quad \cos(0.75)-0.75\approx -0.0183$
Refine the estimate
Checking more closely gives a crossing near $0.739$ . At that input, the two outputs agree to about three decimal places. The graph or a calculator gives an approximate answer, not an exact algebraic value.
$x\approx 0.739$
Read the inequality
The difference is positive to the left of the crossing on the stated interval, so cosine is greater than the input there. At the crossing the values are equal, and the strict inequality excludes it. The left endpoint is included because the inequality is true there.
$\cos x>x\quad\text{for}\quad 0\leq x<0.739\text{ approximately}$

Answer: The equation has a solution at approximately

x=0.739

. The inequality holds approximately for

0\leq x<0.739

.

Check: At

x=0

,

\cos 0=1>0

, so the inequality includes the left endpoint. At the crossing, the two sides are equal, so that point is excluded from the strict inequality.

Common mistakes and how to avoid them

Reporting the vertical coordinate of an intersection as the solution.

Correction: The solution to an equation in

x

is the horizontal coordinate, because it is the input where the outputs match.

Using a rounded crossing value as though it made the two sides exactly equal.

Correction: Describe a numerical answer as approximate and use a suitable approximation symbol.

Including the equality point in a strict inequality solution.

Correction: For

<

or

>

, exclude any input where the two sides are equal.

Using degree mode when the question specifies radians.

Correction: Check the angle unit before evaluating a trigonometric function.

Lesson summary

Solve a non-algebraic equation by finding where the graphs of its two sides intersect.
Use numerical values to narrow down and refine an approximate solution.
For an inequality, compare which graph is higher or check the sign of the difference.
Check the domain, interval, angle units, and whether endpoints are included.

Check your understanding

Question 1

A graph of

y=f(x)

and

y=g(x)

crosses at

x=2.4

. What does this tell you?

The equation $f(x)=g(x)$ has a solution at approximately $x=2.4$ .
The equation has the solution $y=2.4$ .
The inequality $f(x)>g(x)$ is true for every input.
The graphs do not share an output.

Show answer and explanation

The equation

f(x)=g(x)

has a solution at approximately

x=2.4

.

At the crossing, the outputs are equal. The input coordinate,

2.4

, is the solution to the equation.

Question 2

On an interval, a graph of

f

is below a graph of

g

. Which comparison is true there?

$f(x)>g(x)$
$f(x)<g(x)$
$f(x)=g(x)$ at every input
No comparison can be made from the graph

Show answer and explanation

f(x)<g(x)

A lower graph has smaller output values, so

f(x)<g(x)

on that interval.

Key terms

Domain: The set of input values allowed for a function or problem.
Intersection: A point where two graphs meet and have the same coordinates.
Numerical solution: An approximate input value found using calculations or a graph.
Radians: A unit for measuring angles used by trigonometric functions in this lesson.

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D3.2 · Solve non-algebraic equations and inequalities