Linear Equations and
Inequalities
Much education today is monumentally
ineffective. All too often we are giving
young people cut flowers when we
should be teaching them to grow their
own plants.
John W. Gardner
Algebra is great fun - you get
to solve puzzles!
A Puzzle
What is the missing number?
–2=4
Algebra
Well, in Algebra we don't use blank boxes,
we use a letter (usually an x or y, but any
letter is fine). So we would write:
x–2=4
x–2=4
It is really that simple. The letter (in this
case an x) just means "we don't know this
yet", and is often called the unknown or
the variable.
And when you solve it you write:
x=6
How to Solve
Algebra is just like a puzzle where you
start with something like:
x–2=4
and you want to end up with something
like:
x=6
How to Solve
Basic approach to solve:



Work out what to remove to get "x = ..."
Remove it by doing the opposite
(adding is the opposite of subtracting)
Do that to both sides
How to Solve
To remove it, do
the opposite, in
this case add 2:
Do it to both sides:
Which is:
We want to
remove the “-2"
x – 2 = 4
+2 +2
0
6
x – 0 = 6
x = 6
Why did we add 2 to both
sides?
To "keep the balance“
Add 2 to left side
Out of balance
Add 2 to right
side
Back in balance
Algebra-Balance Scale
Solve simple linear equations using a balance
beam representation.
3x + 3 = x + 7
Algebra-Balance Scale
Add tiles representing variables and constants to
both sides of the scale to balance the equality.
3x + 3 = x + 7
Algebra-Balance Scale
What happens if we move a number 1 from left
hand side
3x + 3 = x + 7
No longer balanced……… 3x + 2 does NOT equal x + 7
What do we need to do to make it balance?
Algebra-Balance Scale
We need to move 1 from the other side to get a
new balance equation
3x + 2 = x + 6
We have balance again since 3x + 2 does EQUAL x + 6
Algebra-Balance Scale
Let’s remove 2 from both sides to eliminate the
constant from one side of the equation
3x
=x+4
We don’t have numbers on both sides; but we
do have x’s
Algebra-Balance Scale
Now let’s remove the “x” from the right hand
side of the equation
3x
=x+4
Taking x just from one side caused imbalance
We must take x from other side
Algebra-Balance Scale
Now we have balance with 2x on one side and 4
on the other side
2x
=
4
Since twice a number is 4, then
x = 2 ………….wow easy
Algebra-Balance Scale
3x + 3 = x + 7
3x + 3 - 3 = x + 7 - 3
3x = x + 4
3x -x = x - x + 4
2x = 4
x=2
Subtract 3
from both
sides
Subtract x
from both
sides
Divide by 2
from both
sides
Solve: r + 16 = -7
To solve, you must get the variable by itself.
What number is on the same side as r ?
16
To get r by itself, we must undo the “add 16”.
What is the opposite of addition?
Subtract 16
1.
2.
3.
4.
Draw “the river” to
separate the
equation into 2
sides
Subtract 16 from
both sides
Simplify vertically
Check your answer
by substituting your
answer back into
the problem
r + 16 = -7
- 16 -16
r
= -23
-23 + 16 = -7
Solve: x + 2 = -3
Get the variable by itself. What is
your first step?
1.
2.
3.
4.
Add 2 to both sides
Subtract 2 from both sides
Add 3 to both sides
Subtract 3 from both sides
To solve two-step equations, undo
the operations by working backwards.
Example:
x
 3  7
2
Ask yourself,
1. What is the first
thing we are doing
to x?
2. What is the second
thing?
Recall the order of
operations as you
answer these
questions.
• Dividing by 2
• Subtracting 3
To undo these steps, do the opposite operations in opposite order.
Use a DO-UNDO chart as a shortcut to
answering the questions.
In the table, write the opposite operations in the opposite
order
x
 3  7
DO UNDO
÷2 + 3
- 3 ×2
Follow the steps
in the ‘undo’
column to
isolate the
variable.
2
1.
2.
3.
4.
5.
6.
Draw “the river”
Add 3 to both sides
Simplify
2×
Clear the fraction -Multiply
both sides by 2
Simplify
Check your answer
+3
x
2
8
+3
= - 4×2
x = -8
 3  7
2
-4 – 3 = -7
Solve:
1.
2.
3.
4.
5.
6.
Draw “the river”
Add 1 to both
sides
Simplify
Divide both sides
by 2
Simplify
Check your
answer
2x - 1 = -3
+1 +1
2x = -2
2
2
x = -1
2(-1) - 1 = -3
-2 – 1 = -3
D U
2 +1
-1 ÷2
×
Solve
d  4
2
1. Draw “the river”
2. Clear the fraction
- Multiply both
sides by 2
3. Simplify
4. Add 4 to both
sides
5. Simplify
6. Check your
answer
2
d 4
D
3
3
-4
÷2
2
2
d–4= 6
+4 +4
d = 10
10  4
2
3
U
2
+4
×
Solve
d 1
3
1.
2.
3.
4.
d = -7
d = -19
d = -17
d = 17
 6
Equivalent Equations
Equations with the same solutions are called equivalent
equations.
The Addition Principle
For any real numbers a, b, and c,
a = b is equivalent to a + c = b + c.
The Multiplication Principle
For any real numbers a, b, and c with c  0,
a = b is equivalent to a • c = b • c.
Solve: 2 x  5  11
Solve:
3
2
Solution
x  5  11
3
2
3
x  5  5  11  5
2
Adding 5 to both sides
(Addition Principle)
x  16
3
3 2
3
 x  16 
2 3
2
1 x 
8 2 3
2
x  24
Multiplying both sides by 3/2
(Multiplication Principle)
Simplifying
Solve.
4
(6 x  1)  8
5
Solution:
4
(6 x  1)  8
5
5 4
5
 (6 x  1)  8 
4 5
4
6 x  1  10
6 x  1  1  10  1
6x  9
6x

6
x 
9
6
3
2
An Equation-Solving Procedure
1. Use the multiplication principle to clear any
fractions or decimals. (This is optional, but can ease
computations.
2. If necessary, use the distributive law to remove
parentheses. Then combine like terms on each side.
3. Use the addition principle, as needed, to isolate all
variable terms on one side. Then combine like terms.
4. Multiply or divide to solve for the variable, using
the multiplication principle.
5. Check all possible solutions in the original equation.
Problem Solving Strategy
1. Familiarize yourself with the problem (draw
pictures if applicable).
2. Translate to mathematical language. (This often
means writing an equation.)
3. Carry out some mathematical manipulation.
(This often means solving an equation.)
4. Check your possible answer in the original
problem.
5. State the answer clearly, using a complete English
sentence.
The apartments in Wanda’s apartment
house are consecutively numbered on
each floor. The sum of her number and her
next door neighbor’s number is 723. What
are the two numbers?
Solution
1. Familiarize. The apartment numbers are
consecutive integers.
Let x = Wanda’s apartment
Let x + 1 = neighbor’s apartment
2. Translate.
Rewording:
Translating:
3. Carry out.
First integer
plus second integer is 723



x

 x  1
x + (x + 1) = 723
2x + 1 = 723
2x = 722
x = 361
If x is 361, then x + 1 is 362.


 723
4. Check. Our possible answers are 361 and 362.
These are consecutive integers and the sum is
723.
5. State. The apartment numbers are 361 and 362.
Digicon prints digital photos for $0.12 each plus
$3.29 shipping and handling. Your weekly
budget for the school yearbook is $22.00. How
many prints can you have made if you have
$22.00?
Solution
1. Familiarize. Suppose the yearbook staff takes 100
digital photos. Then the cost to print them would be the
shipping charge plus $0.12 times 100.
$3.29 + $0.12(100) which is $15.29.
Our guess of 100 is too small, but we have familiarized
ourselves with the way in which the calculation is made.
2. Translate.
Rewording:
S hipping
plus
Translating:

$3.29


3. Carry out.
photo cost

0.12( x )
is


$22

22
3 .2 9  0 .1 2 x  2 2
0 .1 2 x  1 8 .7 1
x  1 5 5 .9  1 5 5
4. Check. Check in the original problem. $3.29 + 155(0.12) =
$21.89, which is less than $22.00.
5. State. The yearbook staff can have 155 photos printed per
week.
Solving Inequalities
●Solving
inequalities follows the same
procedures as solving equations.
●There
are a few special things to
consider with inequalities:
●We need to look carefully at the inequality sign.
●We also need to graph the solution set.
How to graph the solutions
> Graph any number greater than. . .
open circle, line to the right
< Graph any number less than. . .
open circle, line to the left
 Graph any number greater than or equal to. . .
closed circle, line to the right
Graph
any
number
less
than
or
equal
to.
.
.

closed circle, line to the left
Solutions of Inequalities
An inequality is a number sentence containing >
(is greater than), < (is less than),  (is greater
than or equal to), or  (is less than or equal to).
Determine whether the given number is a
solution of x < 5: a) 4 b) 6
Solution
a)
b)
-4
-3
-2
-1
0
1
2
3
4
5
6
Since 4 < 5 is true, 4 is a solution.
Since 6 < 5 is false, 6 is not a solution.
b) y  4;
Graph each inequality: a) x < 3,
c) 3 < x  5
Solution
a) The solutions of x < 3 are those numbers less
than 3.
Shade all points to the left of 3.
The open dot at 3 and the shading to the left
indicate that 3 is not part of the graph, but
numbers like 1 and 2 are.
-4
-3
-2
-1
0
1
2
3
4
5
6
b) The solutions of y  4 are shown on the number
line by shading the point for 4 and all points to the
right of 4. The closed dot at 4 indicates that 4 is
part of the graph.
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
c) The inequality 3 < x  5 is read “3 is less than x
and x is less than or equal to 5.”
-5
-4
-3
-2
-1
0
1
2
3
4
5
The Addition Principle for
Inequalities
For any real numbers a, b, and c:
a < b is equivalent to a + c < b + c;
a  b is equivalent to a + c  b + c;
a > b is equivalent to a + c > b + c;
a  b is equivalent to a + c  b + c.
Solve x + 6 > 2 and then graph the
solution.
Solution
x+6>2
x+66>26
x > 4
-5
-4
-3
-2
-1
0
1
2
3
Any number greater than 4 makes the
statement true.
4
5
Solve 4x  1  x  10
Solution
4x  1  x  10
4x  1 + 1  x  10 + 1
4x  x  9
4x  x  x  x  9
3x  9
x  3
-12
-11
-10
-9
-8
-7
-6
-5
-4
-3
Adding 1 to both sides
Simplifying
Subtracting x from both sides
Simplifying
Dividing both sides by 3
-2
-1
0
1
2
The Multiplication Principle for
Inequalities
For any real numbers a and b, and for any positive
number c:
a < b is equivalent to ac < bc, and
a > b is equivalent to ac > bc.
For any real numbers a and b, and for any negative
number c:
a < b is equivalent to ac > bc, and
a > b is equivalent to ac < bc.
Similar statements hold for  and .
Solve and graph each inequality:
a) 1 x  4
b) 4y < 20
7
Solution
a) 1 x  4
7
4
1
x  74
Multiplying both sides by 4
x  28
Simplifying
3
.
5
10
15
20
25
30
b) 4y < 20
4 y
4

20
Dividing both sides by 4
4
At this step, we reverse the
inequality, because 4 is negative.
y  5
-6
-5
-4
-3
-2
-1
0
1
2
3
4