3.3
Graphical
Solutions of Equations and Inequalities
Homework: Every Other Odd 1-37, All Odd 49-53
A. Solve equations graphically.
1. Graph each side of the equation as 
, respectively.
2. The x-coordinate of the point of
intersection is the solution of the equation.
Ex.
Solve graphically for the equation 12x + 1 = 7 – 3x
Enter each side of
the equation into calculator as a function in Y= window.
To see graph in
the Standard Viewing window, hit ZOOM, 6. 
To find the point
of intersection, enter 2nd CALC, 5.
When you see
prompts, First curve? Hit ENTER; Second curve? ENTER, and Guess? ENTER.
The point of intersection is given. The x-coordinate of this point is the
solution for the equation 12x + 1 = 7 – 3x.
The solution
is .4 , or written as a fraction
2/5.
Ex.
Solve graphically for the equation
x – 7 = 3x – 3
Enter each side of
the equation into calculator as a function in Y= window.
To see graph in
the Standard Viewing window, hit ZOOM, 6.
To get a better view of the intersection, we need to see farther down. Go to
WINDOW, and decrease the y-min from the standard setting of –10 to –20. Then
hit GRAPH.
To find the point
of intersection, enter 2nd CALC, 5.
When you see
prompts, First curve? Hit ENTER; Second curve? ENTER, and Guess? ENTER.
The solution for the equation is –2.
(Note: Since we
changed the WINDOW settings, be sure the next time you graph to use ZOOM, 6 to
return to standard viewing window.)
Practice Problem:
Solve graphically
for the equation 2x – 5 = - x + 7.
B. Types of equations (3).
1. Identity –
Always true
Example: 3( x + 4 ) = 3x + 12
You may notice
that the expression on the left side of the equation is always equal to the
expression on the right.
What
happens when you graph:
You see that since the functions are equal, the graph of each is represented by
the same line.
Graphically,
solutions are found at the point(s) of intersection. Since these graphs
coincide at EVERY point, the equation is an identity and has infinite
solutions. The set of solutions for x is the set of all real numbers, which can
be represented by the symbol 
.
What happens when you solve algebraically:
3( x + 4 ) = 3x +
12
3x + 12 = 3x +
12 Removed parentheses.
3x – 3x + 12 = 3x – 3x +
12 Subtracted 3x from both sides of
equation.
12 = 12 Always true.
Solution set is 
.
2. Contradiction
– Never true
Example: x + 6 = x + 1
What happens when you graph:
Graphically, solutions are found at the point(s) of intersection. Since these
graphs are parallel lines and therefore, NEVER INTERSECT, the equation is a
contradiction and has no solution. The set of solutions for x is the empty set,
which can be represented by the symbol 
.
What happens when you solve algebraically:
x + 6 = x + 1
x – x + 6 = x – x +
1 Subtracted x from both sides of the
equation.
6 = 1 Never true
Solution set is 
.
3. Conditional
– True only for a specific value
Example: 2x + 7 = 3x – 10
What
happens when you graph:
Change WINDOW as follows:
GRAPH
Find Intersection
by using 2nd CALC, 5. First curve? ENTER. Second curve? ENTER, and
Guess? ENTER.
The solution for the equation is 17. The equation is a conditional equation, true only for the specific value of 17. The
solution set contains the one element 17, which is written as 
.
What
happens when you solve algebraically:
2x + 7 = 3x – 10
2x – 2x + 7 = 3x – 2x –
10 Subtracted 2x from both sides of the
equation.
7 = x – 10 Combined like terms on each side of the
equation.
7 + 10 = x – 10 + 10 Added 10 to both sides of the equation.
17 = x.
Solution set is 
.
Practice Problem:
Decide if the
equation 9x – 2(x +4) = 7 + 7x – 15 is
a conditional equation, a contradiction, or an identity; and give the solution
set.
C. Solving
inequalities graphically.
1. Graph each side of the inequality as
, respectively.
2. Graph 
= the comparison of 
.
3. Find the x-coordinate of the point
of intersection.
4. Write the inequality statement which
represents the graph of 
.
Ex.
Solve graphically for the inequality 16 – 7x 
10x – 4
Enter each side of the inequality and 
as follows:
To enter the third function, use the following steps. Hit VARS key, use right
arrow to move to Y-VARS, select 1: Function, select 1. Then 2ndMATH,
select 4 for the inequality symbol given in the problem. Then VARS, use right
arrow to move to Y-VARS, select 1: Function, select 2.
GRAPH in standard
window, using ZOOM 6.
The line slanted to the left is the graph of 16 – 7x, the line slanted to the
right is the graph of 10x – 4, and the horizontal line is the graph of the
solutions for 16 – 7x 
10x – 4.
Find the point of
intersection, to find the x-coordinate of the intersection.
We see that the solution set lies to the left of the point of intersection,
which gives the inequality statement.
x 
1.176 (rounding to three decimal places).
Ex.
Solve graphically for the inequality
x + 6 > -2x - 9
Enter each side of
the inequality and 
as follows:
To enter the third
function, use the following steps. Hit VARS key, use right arrow to move to
Y-VARS, select 1: Function, select 1. Then 2ndMATH, select 3 for the
inequality symbol given in the problem. Then VARS, use right arrow to move to
Y-VARS, select 1: Function, select 2.
GRAPH in standard
window, using ZOOM 6.
Find the point of intersection, to find the x-coordinate of the intersection.
We see that the solution set lies to the right of the point of intersection,
which gives the inequality statement.
x > -5 .
Practice Problem:
Solve graphically
for the inequality 2x – 1 > - x – 8.