Download Slide 1

An equation is a mathematical statement that two expressions
are equivalent. The solution set of an equation is the value
or values of the variable that make the equation true. A linear
equation in one variable can be written in the form ax = b,
where a and b are constants and a ≠ 0.
Linear Equations in One
variable
4x = 8
3x –
= –9
2x – 5 = 0.1x +2
Nonlinear
Equations
+ 1 = 32
+ 1 = 41
3 – 2x = –5
Notice that the variable in a linear equation is not under a radical
sign and is not raised to a power other than 1. The variable is
also not an exponent and is not in a denominator.
Solving a linear equation requires isolating the variable on one
side of the equation by using the properties of equality.
To isolate the variable, perform the inverse or opposite of every operation in the
equation on both sides of the equation. Do inverse operations in the reverse order
of operations.
Ex 1: The local phone company charges $12.95 a month
for the first 200 of air time, plus $0.07 for each
additional minute. If Nina’s bill for the month was
$14.56, how many additional minutes did she
use?
monthly
charge plus
12.95
+
additional
minute
charge
0.07
times
*
m
number of
additional
minutes
=
=
total
charge
14.56
12.95 + 0.07m = 14.56
–12.95
–12.95
0.07m =
0.07
m = 23
1.61
0.07
Nina used 23
additional minutes.
Ex 2: Solve 4(m + 12) = –36
Method 1
4(m + 12) = –36
4
4
m + 12 = –9
–12 –12
m = –21
Method 2
4m + 48 = –36
–48 –48
4m = –84
4m –84
=
4
4
m = –21
If there are variables on both sides of the equation, (1)
simplify each side. (2) collect all variable terms on one side
and all constants terms on the other side. (3) isolate the
variables as you did in the previous problems.
Ex 3: Solve 3k– 14k + 25 = 2 – 6k – 12.
Simplify each side by combining
–11k + 25 = –6k – 10
like terms.
+11k
+11k
Collect variables on the right side.
25 = 5k – 10
+10
+ 10
35 = 5k
5
5
7=k
Add.
Collect constants on the left side.
Isolate the variable.
You have solved equations that have a single solution.
Equations may also have infinitely many solutions or no
solution.
An equation that is true for all values of the variable, such as
x = x, is an identity. An equation that has no solutions, such
as 3 = 5, is a contradiction because there are no values
that make it true.
Ex 4:
Solve 3v – 9 – 4v = –(5 + v). Solve 2(x – 6) = –5x – 12 + 7x.
3v – 9 – 4v = –(5 + v)
2(x – 6) = –5x – 12 + 7x
–9 – v = –5 – v
2x – 12 = 2x – 12
+v
+v
–2x
–2x
–9 ≠ –5
x
–12 = –12 
The equation has no solution.
The solution set is the empty
set, which is represented by the The solutions set is all real
number, or .
symbol .
An inequality is a statement that compares two expressions by
using the symbols <, >, ≤, ≥, or ≠. The graph of an inequality is
the solution set, the set of all points on the number line that satisfy
the inequality.
The properties of equality are true for inequalities, with one
important difference. If you multiply or divide both sides by a
negative number, you must reverse the inequality symbol.
These properties also apply to inequalities expressed with >, ≥, and ≤.
Example 5: Solving Inequalities
Solve and graph 8a –2 ≥ 13a + 8.
8a – 2 ≥ 13a + 8
–13a
–13a
–5a – 2 ≥ 8
+2
+2
–5a ≥ 10
–5a ≤ 10
–5
–5
a ≤ –2
Subtract 13a from both sides.
Add 2 to both sides.
Divide both sides by –5 and
reverse the inequality.
Example 5 Continued
Solve and graph 8a – 2 ≥ 13a + 8.
Check Test values in
the original inequality.
Test x = –4
•
–10 –9
Test x = –2
–8 –7 –6 –5 –4
–3 –2 –1
Test x = –1
8(–4) – 2 ≥ 13(–4) + 8 8(–2) – 2 ≥ 13(–2) + 8 8(–1) – 2 ≥ 13(–1) + 8
–34 ≥ –44 
So –4 is a
solution.
–18 ≥ –18 
So –2 is a
solution.
–10 ≥ –5 x
So –1 is not
a solution.