Solving Linear Equations and
Inequalities
• Solving algebraically
• Solving graphically
• Solving equations in more than one variable
• Solving linear inequalities
• Solving double inequalities
• Solving absolute value equations
• Applications
• Solving Algebraically
Example: Solve 2x = x – 9
2x – x = x – 9 – x
(get x on 1 side)
x = -9
(simplify)
•
Solving Graphically
Graph left hand side of equation
and right hand side of the equation
and see where the graphs meet.
y = 2x and y = x -9
x = -9 (just need x-value of point)
• Solve 2x + 1 = -x –9
2x + 1 + x = -x –9 + x (x on 1 side)
3x + 1 = -9 (simplify)
3x + 1 – 1 = - 9 – 1 (get x on own)
3x = -10 (simplify)
x = -10
3
Check: Replace x with answer.
2(-10/3) + 1 = -(-10/3) – 9
-20/3 + 1 = 10/3 – 9
-17/3 = -17/3
Both sides equal so answer is
correct.
•
Graph both sides
x = -3.34 (no y-value required)
• Solve 2x = 2x – 3
2x – 2x = 2x – 3 – 2x
0 = -3 (impossible)
So no answer!
•
Graph
Lines parallel so no intersection
•Solve 2(x + 1)=2x + 2
2x + 2 = 2x + 2 (distribute)
2x + 2 – 2x = 2x + 2 – 2x
2 = 2 (always true)
So every x is an answer!
Lines coincide, so they intersect
everywhere. (same line)
Example: Solve algebraically.
7
1 12
3  
2x
2 x
7
1 12
2 x(  3   ) multiply by LCD
2x
2 x
7  6 x  x  24
simplify
7  6 x  6 x  x  24  6 x get x' s on one side
7  7 x  24 simplify
7  24  7 x  24  24 get x on it' s own
 17  7 x simplify
 17 7 x

7
7
 17
x
7
Solve:
x = -2.5
7
1 12
3  
2x
2 x
Words of caution:
Solving graphically will give you answers that are imprecise. If
you want accuracy you need to solve algebraically.
If I ask you to solve an equation I want a precise answer.
However, you can see if you are in the ball park by graphing.
Solving equations in more
than one variable
Solve for C
9
F  C  32
5
9
F  32  C  32  32
5
9
F  32  C
5
5 59
( F  32) 
C
9 95
5( F  32)
C
9
5( F  32)
C
9
Solving Linear Inequalities
A linear inequality is similar to a linear equation except it is
an inequality. Here are some examples of linear inequalities.
2x  3  3
5 x  2( x  5)  2 x  3
x  2  x 1
2 x  43  4( x  2)
Solve the linear inequality and graph on a number line.
3( x  3)  2 x  5
3x  9  2 x  5 Simplify
3x  9  2 x  2 x  5  2 x Get x' s on 1 side
x  9  5 Simplify
x  9  9  5  9 Get x on it' s own
x  14 Solved
-14
(-, -14) in interval notation
Solve the inequality and graph the solution on the number line.
 3 x  2  11
 3 x  2  2  11  2
 3x  9
 3x 9

divided by a negative flip inequality
3 3
x  3
-3
( -, -3 ] in interval notation
Solve the inequality and graph solution on number line.
1
(9 x  27)  5
3
 3x  9  5
 3x  14
 14
x
divided by negative so flip inequality
3
2
x  4
3
4
2
3
[ -4 2/3,  ) in interval notation
Solving a Double Inequality
Solve and graph on a number line.
 5  x  5  25
 5  5  x  5  5  25  5 trying to get x by itself
 10  x  20
so we take 5 away from every part.
-10
20
( -10, 20 ]
Solving Absolute Value Equations
Solve this absolute value equation.
x 5
Almost everyone has a hard time with these equations. The most
common error is to only give one solution. When in fact there are
usually two answers. Let’s try solving these graphically first. We
will graph the left and right hand sides of the equation and see
where the graphs meet.
y  x and y  5
y=|5|
y=5
x=-5
x=5
Because the graph of an absolute value function is generally a
‘V’, there is a good chance that you will get two answers.
Solving Absolute Value Equations
Algebraically
Solve the absolute value equation.
x 7
Break equation into two pieces
The  piece
x7
or
The - piece
x7
x  7
2 x  10  12
 piece
- piece
2 x  10  12
2 x  22
 (2 x  10)  12
 2 x  10  12
x  11
 2x  2
x  1
Solve the following inequality
5t  6  2t  9
 piece
- piece
5t  6  2t  9
5t  2t  3
3t  3
t  1
 (5t  6)  2t  9
 5t  6  2t  9
6  7t  9
15  7t
15
t
7
Solving Absolute Inequalities Graphically
Solve the absolute inequality
graphically.
x 1  5
The absolute value
function is larger than
y = 5 when x is >= 4 and
x <= -6
or ( - , -6 ]  [ 4,  )
-6
4
Solve the absolute inequality graphically
2x  4  x
y = | 2x – 4 |
x=4
y=x
x  4/3
The absolute value function is less than the x function when
x is less than 4 and greater than 4/3
4/3
4
or ( 4/3, 4 )
Now let’s solve algebraically
x8  5
 piece
- piece
x8  5
x  3
 ( x  8)  5
 x 8  5
 x  13
x  13
-13
-3
So x must be
less than –3 and
greater than -13
(-13, -3)
Solve
2x  5  x
 piece
2x  5  x
x 5  0
x5
- piece
 (2 x  5)  x
 2x  5  x
 3x  5  0
 3 x  5
5
x
3
Answer in
interval notation:
( - , -5/3 ]  [ 5,  ]
or
5/3
5
Applications
Break Even Analysis
I have decided to go into the business of making custom made
tile top tables. The fixed cost of setting up my business is
$1,000. Each table costs $75 to make. I plan on selling the
tables for $115 each. How many tables will I need to sell in
order to break even?
Solution: There are two functions here. A cost function C(x)
and a revenue function R(x) (where x is number of tables
sold.
C(x) = 1000 + 75x
and
R(x) = 115x
The break even point is the point where cost and revenue are
equal. So set the functions equal to each other and solve.
1000 + 75x = 115x
1000 + 75x – 75x = 115x – 75x
1000 = 40x
1000 ÷ 40 = 40x ÷ 40
25 = x
So I will have to sell at least 25 tables to break even.
More on solving graphically
ANSWERS
a. x=-5 & 1 & 5
b. (- , -5]  [ 1, 5 ]
c. (-5,1)  (5,  )
d. (-6,-2)  (2,  )
e. [-4,1]  [6,)
1. f(x) = g(x)
2. f(x) – g(x) = 0
3. f(x) > 0
4. f(x) > g(x)
5. g(x) < 0
6. f(x) < g(x)
Solutions:
1-a
4-c
2-a
5-d
3-e
6-b